{ "count": 245, "curves": [ { "id": 12, "curve_key": "1296296795598284266484863302974277781668866621848145181827394481:-47742962588359813275036508446728062178987599342510036076588219145857596371879546057522868465129", "ainvs": [ "1", "0", "0", "-27006183241630922218434652145297453784768054621836357954737385", "55258058551342376475736699591118191821521067032535079608372404779149413277716173425636721497" ], "rank_lower_bound": 29, "naive_height": 436.0124906483259, "faltings_height": 34.235440688657754, "conductor": "188691278153039213628440510802703336227022485808859170041743892297765880547867276243424847942033952274950770795963060482541371967126966027206485174370", "bad_primes": [ "2", "3", "5", "7", "11", "13", "17", "31", "41", "43", "61", "233", "241", "4139", "678146849364709860535420504397393", "159788990966780131363155786084695062643236502969", "4402149008473369392540402625019227412319473055901" ], "discriminant": "-58514056884179895803252795545068205623215359270190478217393508483140281138663769634024489915971980794469429246265449489823479442097938729025702174957426421151737866434992604323840000000", "regulator": "1433744182671713097629179252379019849.49384267391518181639888988960560090198905327978618140977503", "points": [ [ "2891195474228537189458255536634", "1159930748096124706459835910727318679593425283" ], [ "3402542165322127811451484642234", "1661508223164691055862657623730465560755290883" ], [ "4298760026558467240422107564794", "4313142249890236204790986787384907722927474563" ], [ "3728756667770947009884455714554", "2530180219584734091116528693531660545660397443" ], [ "5991744132052078230511185130234", "10418901628842034362301273055728300669218858883" ], [ "3236493534632768520540227223034", "1324626796262167243658687198416201825373745283" ], [ "78226686134991174232380689386234", "690394210062759896503429654125516779999512554883" ], [ "11492605643548859374635605140234", "35536316911450952155461624238308456029618940883" ], [ "-5143303362384229804906088118566", "7622356511107986864120352355674305680222368483" ], [ "443985655575065435281568435002", "6584468124388858623214803939643557365635620355" ], [ "-979565018904269680752629749766", "8987348422104537684966706438714038633832170883" ], [ "5184894285212178249566461261834", "7390536788003150201273204464695859875505480483" ], [ "-4469171023687146502067179612166", "9310658892841458934133221137392081403414455683" ], [ "3606405835110925482450522970234", "2183644666981703632482662193390480040127898883" ], [ "16151744576785317732688993162234", "61908882092472338946519909276455831463747210883" ], [ "3573684355943766387962362869754", "2094467155115749424853047283659077805560259203" ], [ "-759376049938858166436491644166", "8679171135458197195914024161800061810952119683" ], [ "-5328058719935886182106003119366", "6920588147379497633202935557367499676224350083" ], [ "5380268474895377355583039694554", "8105660240030025092450118297303424395856037443" ], [ "17069233487425098088940203248484", "67583677272795299213867443505411893525786510633" ], [ "5215432542403430758248050783794", "7501515746204716855921710958364078294243814643" ], [ "2838942178046024039763692432122", "1212346280964590308944175800544505700108208003" ], [ "243146882395382015946366404808154/81", "811625272160726332199288136187427505366582108107/729" ], [ "2558229016839511149831260080762", "1706598395830079994387505244133382709649637123" ], [ "2361253942905600810977556672634", "2157503396243552448798851089310708763298766083" ], [ "2678312077644931683114439906234", "1462722361020796436741527433473386115047618883" ], [ "3379397084927230910084852603902", "1608494167359575995485655188349208450365853755" ], [ "3632407730870998917912491355514", "2255654937037700801978158381185619053396712963" ], [ "2428778263277521959543043930234", "1998325023610603606161737305486867803334410883" ] ], "submitter": "David Renshaw", "commentary": "Found by Elkies - Klagsbrun (2024). 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A historical rank ≥ 28 record, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:11:54", "updated_at": "2026-07-02 01:49:25" }, { "id": 67, "curve_key": "2672215049531731291973653104974237283828192744046561:-139952357558889314302863352224099518925635263844664742670201709209325962815825", "ainvs": [ "1", "0", "0", "-55671146865244401916117773020296610079754015500970", "161981895322788558220906653027519611838007321625214218991719656790551905956" ], "rank_lower_bound": 27, "naive_height": 355.2703680721914, "faltings_height": 27.482287272229936, "conductor": "44473459337320533154136751324343533168070733706606064610068741773097621300831711931623980390498751106487256208641524555812214", "bad_primes": [ "2", "3", "7", "13", "19", "23", "29", "43", "97", "1333872817", "622918016143240238107", "4018426744559269274579405699", "461516545359925528236128783904359556661401233959492820033" ], "discriminant": "-292295992467036730780372591065803086322380999542983847983425194363595329691796423905092278238635719085287829796511182852120658943291432879857572200448", "regulator": "70127565624872862829955299377887.957802713297922693486537158452665203606944461202704437006234", "points": [ [ "3767967516008165080365044", "2389736302094908158004099904947501190" ], [ "6870254134405565034404108", "10187524517965617942800545361683736678" ], [ "3887185284020449623939380", "2077020998301366905747533719381033926" ], [ "4704247833799635063001076", "2048360739972031724784820678863578246" ], [ "4126561570009022393013236", "1587663907962563318996362180056025478" ], [ "4589477829219012602846900", "1774818405716699582839388275297252934" ], [ "1744288391661626065189796", "8377495495389391047035879698795823126" ], [ "375965292932773063399988", "11878746522289663117790823052090948358" ], [ "-4430058725939313297140384", "17935065674772418581237173320631279206" ], [ "46029381695079838296565796", "308418721198583803941973238472690797126" ], [ "5015368619774521542769364", "2987769291318668561101046595511063430" ], [ "55141979583089031946559900", "405905110011451276640435700460385551166" ], [ "2703830808220294466353748", "5587793124284970779400186615144247334" ], [ "3412724629872318338319668", "3426156011058008602456511184805561094" ], [ "272723117214107051072886140", "4502171870151657762741942725666306991014" ], [ "4732850534022088572670964", "2124602225002897987491873188898646406" ], [ "19225480790209113087907256", "78725996092378368618479740248297817478" ], [ "5213267756598937117846508", "3666105463387143768198032469471386414" ], [ "-4503215618194252049902522", "17926532987110694852440283715314002874" ], [ "10358928712485769814651816", "26398450763063898266637186797421380678" ], [ "6560446866541184312028656", "8894515448962144734398280820434671978" ], [ "4667249764662401626929236", "1954092716090144351072720616414325286" ], [ "3131745787384349113625300", "4283649283716227803355987840842617734" ], [ "243907731994687263474127628", "3807478665185691587984635270031859346574" ], [ "110171466072672245507182388", "1153803508275547153736593941741941166854" ], [ "2631452133741740392491152", "5805818938673165314161211507146370634" ], [ "2398961346477899287733092916", "117498623151243646059583140149253976390406" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Andrej Dujella rank-record table \"Rank >= 27\", Elkies (2016). The page lists this curve y^2 + xy = x^3 - 55671146865244401916117773020296610079754015500970*x + 161981895322788558220906653027519611838007321625214218991719656790551905956 and the 27 independent points submitted here.", "created_at": "2026-05-29 04:46:51", "updated_at": "2026-07-02 01:49:26" }, { "id": 66, "curve_key": "13035271910468252150644679736976114920264729121:-1445924176167521752672574624080130182981403350858921203020924149845169", "ainvs": [ "1", "0", "0", "-271568164801421919805097494520335727505515190", "1673523352045742769296938739782713519216640490554763586630258973092" ], "rank_lower_bound": 26, "naive_height": 318.5519642757846, "faltings_height": 24.458470810082996, "conductor": "440755853609146510472634852930653385420305551113633650634939171542380165741313856027971752379908017654932910", "bad_primes": [ "2", "3", "5", "7", "11", "13", "37", "61", "109", "3180882909509", "101223837495550397872963", "185290954722549979010823774145698082035847060397002762839016907" ], "discriminant": "71895080707631236566227414257088729466603817414726942045972724673173251962604021806427843161273966232348946866110784725884544768000000", "regulator": "64414287721104130201143827656.9267422425825753392963916482410113025800545012498701248313219", "points": [ [ "8171422342464806465484", "-6968834778825385043474309573382" ], [ "15506445321751452148524", "-1091324356491763240297753031369862" ], [ "11039908252424365794924", "-144827641060783314905542709878662" ], [ "-1124601528095626940916", "-1406238603313519698489557881370502" ], [ "15979026151674771866124", "-1189134868428022585423319170045062" ], [ "11497067226620401106124", "-266453792604568993030594792645062" ], [ "7964508116120761886124", "-125826249902394562842770754145062" ], [ "2057879788890780254844", "-1059897903660876637259757866892102" ], [ "11013564232941423844134", "-136087530861067930848004367296452" ], [ "10829312030749636504524", "-51221935741228061266466491121862" ], [ "5848751988213230056404", "-534099270092912134067240836612782" ], [ "11638411535858522295324", "-298922500066105927047772771275462" ], [ "6868334166980311706124", "-363743509637318702264451883645062" ], [ "7381103602441056808764", "-266792452337957421936180167447622" ], [ "-18419992393365020549076", "-652667554475023102928617867030662" ], [ "5493669297710229516324", "-589422439885808591350951805382462" ], [ "10874362643126096194524", "-79405280481739553235221538851862" ], [ "12784040435352607621644", "-539538954823537014427509667959942" ], [ "5007446790086282822724", "-662736416997155792108337796316262" ], [ "36311702775006993614724", "-6300067037234134629601408866834462" ], [ "7959119841967332599004", "-127557175274976008347318287354822" ], [ "-12920905145755852965276", "-1739335347799715861327567809614462" ], [ "-17861114371778492133876", "-908840632427321863500516490045062" ], [ "19507248497293445799354", "-1949133775462458912736164210505272" ], [ "-17316278094075829267956", "-1087992116843026234002459666650502" ], [ "33532502811165086659374", "-5502006283885945731489116619434562" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Andrej Dujella rank-record table \"Rank >= 26\", Elkies (2006). The page lists this curve y^2 + xy = x^3 - 271568164801421919805097494520335727505515190*x + 1673523352045742769296938739782713519216640490554763586630258973092 and the 26 independent points submitted here.", "created_at": "2026-05-29 04:46:23", "updated_at": "2026-07-01 22:29:38" }, { "id": 65, "curve_key": "58683989082049435979413878577717203009129601:-452707874380980964751123855943852496111823697994336414045572254401", "ainvs": [ "1", "0", "0", "-1222583105876029916237789137035775062690200", "523967447200209449943328506898413682821590945806099349816040000" ], "rank_lower_bound": 25, "naive_height": 302.35621581434873, "faltings_height": 23.045960193031338, "conductor": "3421625153333343825198002959770328244842456471613462044687992344664286334910845403447307652180335890", "bad_primes": [ "2", "3", "5", "7", "11", "13", "157", "283", "72551", "895613", "39466373175449369221626360280383658386480681928740709773906484427935361743867171" ], "discriminant": "-1648077180048519919196851388268507997232089165506048976474499224670899793965899322864689351309245989194390271909250018304000000", "regulator": "724084252210012759383791369.1859440608064513839867489159651105029300072050018474263574137", "points": [ [ "641919671491882025200", "1917622583611100541324074905000" ], [ "594994690844383471600", "2678847964182260740553494589800" ], [ "208196841608040479152", "16686940720261011217265226309352" ], [ "-740361154329309511040", "-31989105845069848078376802650600" ], [ "804899774308127940400", "7834276239548635107854922077800" ], [ "77301598762979192560", "20734554180656765176390619976040" ], [ "702911299694560038040", "-3449260992056554312601103761600" ], [ "915485567326094606800", "-13114547973847689345711009896600" ], [ "-26293910519288470100", "-23581682704464936496514064216200" ], [ "709908819594746168740", "-3717262868674905037217938483100" ], [ "875280814115630686150", "11154863867288165850338319192550" ], [ "1643784276034119529840", "54367762877007597564101217327400" ], [ "2310209163245804946160", "100146314980252246302935050670440" ], [ "1909368489723249677080", "-71767455269710117770413293033280" ], [ "673388810420663671600", "-2458352439913512016092058261400" ], [ "-1225141077322557861200", "13524179089651555542230916989800" ], [ "641884454793213323650", "1917498326574315724474707751300" ], [ "1779583257516825231520", "63119495622551336426454649548280" ], [ "1336811684405040135250", "35757154248294043850600009373700" ], [ "591774410894432534320", "-2776985009258775118824482266520" ], [ "4566516049692282828100", "300278030461559541447358362839800" ], [ "-499965890434284752520", "-31784318126596227394232274064800" ], [ "1239030190644321769050", "-30187812602932900034742078815100" ], [ "3417029330608564213375/4", "-81177702701038029224605519995775/8" ], [ "9868611464482118831434/25", "1267809878380229048466900915382388/125" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Andrej Dujella rank-record table \"Rank >= 25\", Elkies (2006). The page lists this curve y^2 + xy = x^3 - 1222583105876029916237789137035775062690200*x + 523967447200209449943328506898413682821590945806099349816040000 and the 25 independent points submitted here.", "created_at": "2026-05-29 04:45:51", "updated_at": "2026-07-01 22:29:39" }, { "id": 10, "curve_key": "5761911457775627774569661721176006225929:-435650393506962818897975621667886917582013897958359471999333", "ainvs": [ "1", "0", "1", "-120039822036992245303534619191166796374", "504224992484910670010801799168082726759443756222911415116" ], "rank_lower_bound": 24, "naive_height": 274.6562636902012, "faltings_height": 20.714078599997542, "conductor": "325763846420202806803685538805031963849862314963649942610049440603050996133862755155678267754410", "bad_primes": [ "2", "3", "5", "11", "13", "17", "29", "31", "41", "458619970494582607679296750333015081", "264240973182971699094661154229360236070105974082503" ], "discriminant": "869228312577660772590804276974894693796770087691229179004716529760427780748163030521421801352204521776094281605900", "regulator": "104899016884108750894133329.51301636361085332121321081420962914892371133491485632339948", "points": [ [ "2005024558054813068", "-16480371588343085108234888252" ], [ "-4690836759490453344", "-31049883525785801514744524804" ], [ "4700156326649806635", "-6622116250158424945781859743" ], [ "6785546256295273860", "-1456180928830978521107520473" ], [ "6823803569166584943", "-1685950735477175947351774817" ], [ "7788809602110240789", "-6462981622972389783453855713" ], [ "27385442304350994620556", "4531892554281655472841805111276996" ], [ "54284682060285253719/4", "-296608788157989016192182090427/8" ], [ "-94200235260395075139/25", "-3756324603619419619213452459781/125" ], [ "-3463661055331841724647/576", "-439033541391867690041114047287793/13824" ], [ "-6684065934033506970637/676", "-473072253066190669804172657192457/17576" ], [ "-956077386192640344198/2209", "-2448326762443096987265907469107661/103823" ], [ "-27067471797013364392578/2809", "-4120976168445115434193886851218259/148877" ], [ "-25538866857137199063309/3721", "-7194962289937471269967128729589169/226981" ], [ "-1026325011760259051894331/108241", "-1000895294067489857736110963003267773/35611289" ], [ "9351361230729481250627334/1366561", "-2869749605748635777475372339306204832/1597509809" ], [ "10100878635879432897339615/1423249", "-5304965776276966451066900941489387801/1697936057" ], [ "11499655868211022625340735/17522596", "-1513435763341541188265230241426826478043/73349586856" ], [ "110352253665081002517811734/21353641", "-461706833308406671405570254542647784288/98675175061" ], [ "414280096426033094143668538257/285204544", "266642138924791310663963499787603019833872421/4816534339072" ], [ "36101712290699828042930087436/4098432361", "-2995258855766764520463389153587111670142292/262377541318859" ], [ "45442463408503524215460183165/5424617104", "-3716041581470144108721590695554670156388869/399533898943808" ], [ "983886013344700707678587482584/141566320009", "-126615818387717930449161625960397605741940953/53264752602346277" ], [ "1124614335716851053281176544216033/152487126016", "-37714203831317877163580088877209977295481388540127/59545612760743936" ] ], "submitter": "David Renshaw", "commentary": "Found by Martin - McMillen (2000). A historical rank ≥ 24 record, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:11:53", "updated_at": "2026-07-02 01:49:27" }, { "id": 9, "curve_key": "924142387616352615747166301588085202681:-28240272628746949088231985451159095195284780253064738305421", "ainvs": [ "1", "0", "1", "-19252966408674012828065964616418441723", "32685500727716376257923347071452044295907443056345614006" ], "rank_lower_bound": 23, "naive_height": 269.176198734463, "faltings_height": 20.268662842613764, "conductor": "113964706497998622564257410134352232583764612126645513242486361984405434455972952111271430", "bad_primes": [ "2", "3", "5", "11", "13", "17", "19", "23", "199", "17858193374841750901974789649", "1006218106634655545344494448610726356220703995276273" ], "discriminant": "-4779639209650129827198370598581502977994781114856713955972654268540889198350896060545934759063721850651300750000", "regulator": "14314346672172296526006093.9515872795609689097642740408996297487603077991878415316140", "points": [ [ "16902136044621724275584661392595/119224493521", "-69455519784971993679807552308609739430858248812/41166906143372569" ], [ "6647882272466103821634772046571/30891226081", "-17137023844710987140049387309945953892946213544/5429411004770479" ], [ "1277229332035649706664846727592/2961427561", "1443380843339272397458721030742392016696304046/161157926442059" ], [ "1754834771916476982132090651/369369961", "49412130720987886904443301152758710388796/7098921280459" ], [ "902743031953703698667092998/307406089", "6538434104009303265024749952830709029353/5389750958437" ], [ "103579510135061476534950819/45091225", "230697883363551870088729854504374414548/302787575875" ], [ "31762044569407766003397375255/14054813809", "1411381089291349753164768808558921002947204/1666240341498377" ], [ "29436984213667648723395/17956", "5657335012046240705357319452802233/2406104" ], [ "1127027270330215920", "3523978127407100674110377602" ], [ "686464244502821899711515/139129", "-394563651945882403580468873435105816/51895117" ], [ "11962675953816366561795/1369", "-1167962768316319592876571517317044/50653" ], [ "30520680805402695175757355/3345241", "-151915114589061403100759698106532333112/6118445789" ], [ "11449775538050756019357635316/967521025", "-1150775031908416918955115365651634494501651/30094741482625" ], [ "4969418243982621661795591770/1285294201", "184569435055535326363669745422918052707327/46079082400051" ], [ "480465113537612829840777315/160801", "-10531550647702714814852169224678207441368/64481201" ], [ "97907154284679777917982542166035/57601436172721", "964874722537391293613786748114488474882993683572/437169613520472565481" ], [ "249989354826313432718977195/4397409", "3941156276776007263792745630379334937996/9221366673" ], [ "54840074123086507808388135/3996001", "387496978790653709721061294119215460988/7988005999" ], [ "9690141319063801580189469420/87590881", "-953144078079942906360903670036536669593542/819763055279" ], [ "882142442406602738753880/76729", "-775394556680837651292166377698874734/21253933" ], [ "2812175950395226936581984/24025", "4712624271973109965160039085789391367/3723875" ], [ "7126269737101017406079752337071371/2947180538019481", "83015454575998684006900205726968222686505350799684/159996363164349841378621" ], [ "2143448685801212487450/841", "10099849221189668277354753748208/24389" ] ], "submitter": "David Renshaw", "commentary": "Found by Martin - McMillen (1998). A historical rank ≥ 23 record, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:11:51", "updated_at": "2026-07-01 22:29:40" }, { "id": 8, "curve_key": "45134376853266785419345109815961449:-9251161693101858048581020605248351326956236733759157", "ainvs": [ "1", "0", "1", "-940299517776391362903023121165864", "10707363070719743033425295515449274534651125011362" ], "rank_lower_bound": 22, "naive_height": 239.38485678351597, "faltings_height": 17.87129795863137, "conductor": "22720638514787473197194583889675055980109503436060704437972911338086049759883790", "bad_primes": [ "2", "3", "5", "7", "13", "17", "37", "47293", "270704849145149791", "60794657878864337775664712674231370427122734380997" ], "discriminant": "3680432205940761304587473756368167508478483727616134375685404923405185646725374901862332174937532900", "regulator": "12992022722213680254541.35531967258530327885791487307445312157323896249403008828342", "points": [ [ "32741153161482344264/3025", "-223089674587110979578532169697/166375" ], [ "215521674613198983365/24649", "-6872949155061353554235704378947/3869893" ], [ "637312541911044643/81", "-1420356190129296832193564087/729" ], [ "-11906250919327880080/361", "-16580788535875788634285886853/6859" ], [ "-136152345735493381/4", "-14482270545045735913281693/8" ], [ "-27830298157016213012252/7134241", "72099692861364392796183359497454267/19055557711" ], [ "4127671322151440", "2626107692045613116291646" ], [ "6175679781777296", "2266254335997033124678449" ], [ "12047255022287093", "1061993236525943920980477" ], [ "416685837455186583191/32761", "5321268222786709669160311587369/5929741" ], [ "149915813139075767108024/10220809", "8704326838108646949177663157917117/32675926373" ], [ "58759417448623559/4", "2030968553150713398654657/8" ], [ "237195157887349854919517/16024009", "-11477798111611307979707215505421441/64144108027" ], [ "9568474434078537574436/687241", "319520556343135681977874272805086/569722789" ], [ "1725892668710258675291/177241", "117378050663464845770966453025039/74618461" ], [ "-35277008506980340471/1024", "48766027143946934186731674507/32768" ], [ "-2752742763529705669/121", "6000532252185982381233585699/1331" ], [ "-18552633109178014", "-4665466215824339436717966" ], [ "-113251707338691187737649969/3304065361", "310152527894831470820009872373229341739/189920981015641" ], [ "-7572001778163591251/729", "-86590661426506799357663502953/19683" ], [ "-380526048554032285152211/11242609", "73081235744931307684790623068490233/37696467977" ], [ "-1503889497722021588110681/42784681", "-160705885170116750151534640924719585/279854598421" ] ], "submitter": "David Renshaw", "commentary": "Found by Fermigier (1997). A historical rank ≥ 22 record, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:11:50", "updated_at": "2026-07-01 22:29:41" }, { "id": 74, "curve_key": "10360501076277308256728157729703672081:16825848144008099204725392608156810861436365523723401479", "ainvs": [ "1", "1", "1", "-215843772422443922015169952702159835", "-19474361277787151947255961435459054151501792241320535" ], "rank_lower_bound": 21, "naive_height": 255.6931918496502, "faltings_height": 19.35836145829672, "conductor": "26112076915897777815571388664430310998157918697219343275140810790098571234096793308930", "bad_primes": [ "2", "3", "5", "7", "13", "17", "23", "47", "4507", "115482611374267602141168398241396608699381902319617225736297616061235976719" ], "discriminant": "479737754043767746536923774462246533556793859277365678757052823009411494064048885744243744500948985175040000", "regulator": "1057662683061657998079887.5644599317446350524828508690354387929045294397158378546", "points": [ [ "800843008889340065933/16", "22662214190910903990783584765347/64" ], [ "10610541066763914590637/2209", "1087744114825178454840094794778034/103823" ], [ "907186946780634143", "728916386168451830641677698" ], [ "196833201085564442194083107/227919409", "2277807398930440819587410184793923763894/3440899317673" ], [ "185463474139064652528000075/366301321", "225699857838583242849473830466481978146/7010640982619" ], [ "-12485261071234691432503/123904", "1543303353428939982282171752702539/43614208" ], [ "-59703014087684747037/361", "741881245094154068525036126962/6859" ], [ "-73270463404799613067/361", "866878137858638792891117943482/6859" ], [ "-360733396398627565", "106985840484096728947883974" ], [ "-389445180957906897", "74288355118790673852542098" ], [ "-1474458350349858512665407/14205361", "2278493401578368084310409028259332632/53540005609" ], [ "-114305856035468892691779277/278589481", "16972779768877136292841029639987095378/4649937027371" ], [ "-21972533600828202797/81", "100790786584963504563876005302/729" ], [ "-25047938415396324842058977/71216721", "68347192566984943007522052612937752062/600997908519" ], [ "3434828081885118352213715284707/5137262501809", "4279912483838925044234939165329697576812433846/11643877735262694377" ], [ "-227656313261676647", "133660024327268949095297798" ], [ "-4098089434105992137835293/12552849", "5660088413991351759301403659890889706/44474744007" ], [ "2657828735869178020212617/1495729", "4174499731549997186596131721273201376/1829276567" ], [ "883965004314243424124994323/850947241", "23250077986002214917145041708721276812178/24822981967211" ], [ "37543938954172817209003/73441", "1224097915991280099903835490020298/19902511" ], [ "19165312347502458410162233/17214201", "75593839815741485450348997055551694952/71421719949" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Nagao-Kouya (1994), “An example of elliptic curve over Q with rank >= 21,” as reproduced in Nagao, “Construction of high-rank elliptic curves.” The paper gives this minimal model y^2 + xy + y = x^3 + x^2 - 215843772422443922015169952702159835*x - 19474361277787151947255961435459054151501792241320535 and 21 independent points. Three point coordinates here use the values from Nagao’s scanned paper rather than the later HTML mirror, whose P2, P12, and P20 appear to contain transcription errors.", "created_at": "2026-06-01 14:26:55", "updated_at": "2026-07-01 22:29:41" }, { "id": 245, "curve_key": "1242259750673056662437665286001:-1388157434854180878311718734402758619153189001", "ainvs": [ "1", "-1", "1", "-25880411472355347134118026792", "1606663697747901005185875883284420820193259" ], "rank_lower_bound": 20, "naive_height": 207.88861293214381, "faltings_height": 15.133214475051384, "conductor": "272066437942638823321634957004224153562929337633497250319389959310", "bad_primes": [ "2", "3", "5", "13", "19", "37", "7770053", "763973980372286963203", "55722582408764114465841769948159" ], "discriminant": "-5737429639444055753512327119829925015713334222025694979826677816527336990079139840000", "regulator": "4687798591445119467.32292703579184309810358227475172994834982074863674547612846", "points": [ [ "45492869957717", "723492619381097668641" ], [ "103561597049417", "192731719560345258741" ], [ "111830641498667", "333174093470012383491" ], [ "316677539205677", "5016853361860584594681" ], [ "92008168616117", "65953506306453213441" ], [ "147881370863967", "-1006695588776782646359" ], [ "213774572035037", "2417332168050319812441" ], [ "81536578719167", "196306066193578043241" ], [ "294423088161917", "-4416891036870168333159" ], [ "227707758469997", "-2742321749648483486439" ], [ "94836710011277", "-72195187870086062919" ], [ "132211123753677", "704277531895181059641" ], [ "91603447091117", "-67755953909555099559" ], [ "60334338604917", "514603898948192002241" ], [ "82129229001677", "-187361656905064421319" ], [ "20079468981897", "-1046467649553400314559" ], [ "107362804572947", "256157490251353593441" ], [ "305606512192501/9", "-23646417837002335990141/27" ], [ "1413476686249589", "-52811301360597927383607" ], [ "3892455620987453/49", "78262410657484049671263/343" ] ], "submitter": "David Renshaw", "commentary": "Rank-20 at naive height 207.89: Mestre-family specialization T=5801/160 found by wide-denominator vectorized sieve (n>20 window), certified via famcert3 with base-point injection.", "created_at": "2026-07-07 01:03:48", "updated_at": "2026-07-07 01:04:58" }, { "id": 243, "curve_key": "37977539895016519929470555599561:-252010427891277533784631384743578099062204450309", "ainvs": [ "1", "0", "1", "-791198747812844165197303241658", "291678735985274857428612896086571996361540568" ], "rank_lower_bound": 20, "naive_height": 218.29159930381655, "faltings_height": 16.146949249509756, "conductor": "123989445321322909065995690886185865732040186216447789402341303782551770", "bad_primes": [ "2", "3", "5", "17", "19", "23", "29", "31", "59", "1657", "204191283231710516050941879612935200616637742318196045393" ], "discriminant": "-5054684863451019643038749147096969353867668542170076589556829874258223258043978504934937500", "regulator": "26922583187358235172.6849204969274253584348869249045696391195036218961397153261", "points": [ [ "436876547159399", "5422665630197378101065" ], [ "375990611505449", "-6881048738091909428565" ], [ "475838366133299", "-4789192526536875389535" ], [ "309626103514289", "8739924684576584646255" ], [ "579770808085271/4", "-107344757768733870289755/8" ], [ "-151670202942496", "20203737650008769711175" ], [ "474836608215689", "4801033336095140780355" ], [ "1399900845934769", "43903256069991280960095" ], [ "413316049389764", "5938892007725126484555" ], [ "-491678658233761", "23702995629264462353655" ], [ "207182831653769", "-11689705199855651038365" ], [ "2826878413762139", "-143684810483260713380445" ], [ "972792119473619", "21037676494215690500235" ], [ "733163671616639", "-10280918119880512036695" ], [ "1159271608488944", "-30535585821516332183880" ], [ "584393010154124", "5374644645235395266835" ], [ "1001929050655889", "-22466688030826232997945" ], [ "382267730323742", "6714852110077627182165" ], [ "467508117994889", "4895637428325170187555" ], [ "3327907052188819", "-185782396222315328318005" ] ], "submitter": "David Renshaw", "commentary": "New curve of rank >= 20 and naive height 218.2916, beating the previous rank-20 record 223.3165 (#92). Found as the specialization T = 3895/6 of the Mestre/Fermigier rank-12 quartic family on the rational Mestre sextuple mestre_ais(u = -7/2, v = 3/2) (roots {-1455/4, 2955/4, 1437/2, -1149/4, -1851/4, -687/2}): q(x,T) = p6(x-T) p6(x+T) = g^2 - r with deg r = 4, curve = Jacobian of y^2 = r(x). Candidate surfaced by a staged Mestre-Nagao sieve (M=1500 stage 1 over T = m/n, m <= 5000, n <= 20; M=6000 rescore ranked it #1 of 3135 candidates under the rank-20 height bar, score 103.65 vs 98.36 for Fermigier's rank-22 curve in the same normalization). The 20 independent points were assembled from hyperellratpoints on the integral quartic model to height 1e7 plus the 12 injected family base points x = ai +/- T and an integer sweep ellratpoints(E, [1e12, 1]); independence certified by positive-definite Neron-Tate Gram matrix (eigenvalue margin 1e-7 at 90 digits).", "created_at": "2026-07-02 02:02:39", "updated_at": "2026-07-02 02:02:39" }, { "id": 92, "curve_key": "212995778885315694793211765486641:-3099031675334352687040090522193313305659658091689", "ainvs": [ "1", "1", "1", "-4437412060110743641525245114305", "3586842216822165612930264910099076801587288127" ], "rank_lower_bound": 20, "naive_height": 223.31647541368378, "faltings_height": 16.4256875981555, "conductor": "2876153493562761211278364526603564191699143885403233935132057708367930", "bad_primes": [ "2", "3", "5", "7", "13", "17", "31", "79", "1049", "71889448247", "40200713707633", "491007790268548705232623905732119" ], "discriminant": "34158080673335280296970759332689207611899474944643988972429089096818399452131514624000000000", "regulator": "14199318900391412470.1467052372488935270092221520193246449884249061865323699427", "points": [ [ "2322567250548927", "76218907815416964881536" ], [ "1159287617040927", "-792075071337327918464" ], [ "3260380495594177", "154198893197321089113536" ], [ "1085033861856627", "7036233563582957274136" ], [ "1057541405086527", "8766013807279112883136" ], [ "2051321472766927", "55821856559912025117536" ], [ "1349853674212677", "-7520658488989426177214" ], [ "-162875228134573", "-65614525056065801511464" ], [ "1973958997485927", "50190953064845966726536" ], [ "1115609187332427", "4989362320694872550536" ], [ "35632774335609831/25", "-1569661495703560782391744/125" ], [ "1065196583526527", "-8291243444403322249664" ], [ "1272177369204427", "-776998561028702009464" ], [ "1577638306201227", "-22646461833194926710164" ], [ "1076965981719177", "7551525239803692957286" ], [ "336801666567727", "-46157552734600369764064" ], [ "1157870843592447", "-1099527333816823674944" ], [ "949628343705087", "15143284705038794180416" ], [ "11740414067153143/9", "114951115455620018279872/27" ], [ "2666485382308185", "103506731053981149881008" ] ], "submitter": "David Renshaw", "commentary": "New curve of rank >= 20 and naive height 223.316, found as the specialization T = 28917/10 (t = 28917/20) of the 2-parameter Mestre rank-12 family over Q(t) used by Fermigier in 'Une courbe elliptique definie sur Q de rang >= 22' (Acta Arith. 82 (1997), sextuple {0,55,314,378,1007,1036}), located by a staged Nagao-sum sieve over ~1.9M specializations; 20 independent points certified by positive-definite Neron-Tate height pairing.", "created_at": "2026-06-11 19:26:11", "updated_at": "2026-07-01 22:29:42" }, { "id": 226, "curve_key": "2559591262338637286657802227947681:-131846891815534571140347404460393717853020072247921", "ainvs": [ "1", "0", "0", "-53324817965388276805370879748910", "152600569230942786227963554343099291511940660100" ], "rank_lower_bound": 20, "naive_height": 230.81145160519077, "faltings_height": 17.12490323263689, "conductor": "282182911611488014589765703157855796043317619561544915217487973922307930", "bad_primes": [ "2", "3", "5", "7", "31", "41", "59", "79", "1615673", "140389069210932330034578246587156827594757780759118714191" ], "discriminant": "-355568117955757642974912655755100242442938536081550074372306129807997878195940027796106400000000", "regulator": "22741880954418501814.3623903478549341506058195963051390323906246906003515483700", "points": [ [ "618601125946370", "346194302434275472478840" ], [ "4274187758343020", "52574304621988571796290" ], [ "1237465552904420", "-297502775169095344544710" ], [ "2942610078779420", "-145486591821952222859710" ], [ "-7399759529700580", "376838035640579185200290" ], [ "4573716239571920", "-66219377082644347492210" ], [ "10682154885034220", "-895489011065482379902510" ], [ "5722894198425740", "-186713018618338452812110" ], [ "2302662274779020", "-204989810208697277535310" ], [ "-6015123623406730", "-505686134502279456010810" ], [ "-913586879332288", "447833570226927052727006" ], [ "3467158396446920", "-96924402816950500267210" ], [ "-2962025209897180", "533443925142244344432890" ], [ "-1020997989620680", "453851050081375324404290" ], [ "6555760311182919080/2209", "-14859538369702735477983935330/103823" ], [ "35422287391675744/9", "1640631393230902228935254/27" ], [ "1647630816630975820/361", "-449530896105195331951787690/6859" ], [ "479223846427162851980/113569", "1996522496958091344388813918370/38272753" ], [ "417628515448521851180/94249", "1666221552406370616415079717270/28934443" ], [ "80652470616459665420/351649", "78130442711385740206728677920930/208527857" ] ], "submitter": "Alexey Pozdnyakov", "commentary": "Generated with assistance from OpenAI Codex, based on GPT-5. Method: Generalized Mestre/Fermigier quartic-family search: Mestre-Nagao stage sieve, high-M rescore, exact-height triage, conductor analysis, and GP famcert certification. Candidate came from family sextuple_C_m138_138_162_m60_m90_m12 with specialization T=10167/350.", "created_at": "2026-06-25 23:18:43", "updated_at": "2026-07-01 22:29:42" }, { "id": 7, "curve_key": "20692463076784016542001380602775969:-4455028991836780152445498634264721662716355492365297", "ainvs": [ "1", "0", "0", "-431092980766333677958362095891166", "5156283555366643659035652799871176909391533088196" ], "rank_lower_bound": 20, "naive_height": 237.8517466226108, "faltings_height": 17.85023793381406, "conductor": "73813242020125452593092920037241715783942979322145629022774016712292126930", "bad_primes": [ "2", "3", "5", "7", "13", "17", "19", "29", "53", "1759", "539449", "1884347", "78324820513", "388882789386500953248084998144029301891" ], "discriminant": "-6358347962741427332351207823555533236599280133229903044072065747055673142684922784840133195530240000", "regulator": "2975173777358668583558.17709962338365235667750391729197206924733709891658896356", "points": [ [ "1117677105220842826524/37249", "31530479477185489011505872316434/7189057" ], [ "38095017214360176", "6634638907482675334232862" ], [ "128263157005359747/4", "39438837388807975937649915/8" ], [ "173541370721241727764/4489", "2045813113492578321709774985406/300763" ], [ "114037038978699019879860444/2903808769", "1093029826650184196976652135696199191086/156477543135103" ], [ "102579683196689625565576980/3236130769", "889405931520755349254783091883555261398/184093771056103" ], [ "520590665688949735068/11881", "10865365165484759274005818215450/1295029" ], [ "201537570874848579/4", "84414630327852273660698571/8" ], [ "8566017671075667672/169", "23408663211165662031648247674/2197" ], [ "84810811649507676", "24054695979596704444705362" ], [ "-21830796739843140", "2040388505636168283880914" ], [ "-2234086367006310516/121", "3476314228926730107073128678/1331" ], [ "-398890292913112314601476/47513449", "936915725382816974616861962133589434/327510203957" ], [ "-38850378311984740900/5041", "1013647136758546790991381788254/357911" ], [ "410916153652874282067804/58874929", "712483344051989593825064319402912354/451747330217" ], [ "3030869760973710007623516/266375041", "5708794986061809828924957672204713682/4347507044161" ], [ "5668123803956059068/361", "10307638984731401904281889030/6859" ], [ "-5809361085179727432048324/267289801", "9018279752358533631568966242553347738/4369920956549" ], [ "-256381598399113962133604/10169721", "1315536690951996543879087852628778/32431240269" ], [ "479228870284501996956/167281", "135888316201098799476616096547298/68417929" ] ], "submitter": "David Renshaw", "commentary": "Found by Nagao (1993). A historical rank ≥ 20 record, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:11:48", "updated_at": "2026-07-01 22:30:39" }, { "id": 40, "curve_key": "11737808320143340870262567384110164924996363543449272977921:-830917597293950141847936654026788953089540835356843717953963823812788614617047247539969", "ainvs": [ "1", "-1", "1", "-244537673336319601463803487168961769270757573821859853707", "961710182053183034546222979258806817743270682028964434238957830989898438151121499931" ], "rank_lower_bound": 20, "naive_height": 401.13049623842045, "faltings_height": 31.462688360849253, "conductor": "14143803817470093613466453371895612334117548667434143819728526037275098905399795800409355189270", "bad_primes": [ "2", "3", "5", "7", "13", "17", "19", "23", "31", "79", "131", "137", "227", "229", "337", "443", "593", "751", "1093", "1297", "2113", "8231", "8629", "19937", "33073", "47419", "157559", "479081", "858259", "2391953", "11332579" ], "discriminant": "536322869068731689907926014223523897218983690620200910078755027869276074296716149126908879546977218241763631376739573056546560007049072199214175253755603358448062903360000", "regulator": "2345864571013562003762503.94206244554266939212953056905157720878181020686213493", "points": [ [ "-5976635286513806621064126789", "-1486518161621811752215471892311622350764326" ], [ "595416388787490259443766591", "903504029373113768509734316584679010931354" ], [ "2434562872293108275107029075", "617088105910990842809058933920186352787402" ], [ "3513074027344435171140978981", "382084415613590558839076708139976548338374" ], [ "399682145249051758133327419", "929535737737305075763667416604801178363754" ], [ "-10714754038296881855524018251", "-1533542840817584957359022971329370697480444" ], [ "-16034220456847626275437501599", "-871976826237835562350053313866844829289386" ], [ "1185828672355214392425799131", "820608248516937017244986013572105379548074" ], [ "-11190697582885409770718510409", "-1515528657208794070421971888182751388116946" ], [ "2634316446310680332042122261", "579484021184216876019568884495793999507014" ], [ "64222149978369055569434725591", "15815822465644868585416444171217848754979354" ], [ "23945425437351916471937562579", "2972551260335618106366053972465493213480034" ], [ "251112537077876411029986504415821/72361", "7660680492845270503732516891131042427064917041476/19465109" ], [ "13094114400583295432756346651", "69052082853368511623611930199368498525114" ], [ "2689776334541089917424552236511", "4411303346731260129651357011579276750542331014" ], [ "-2627014038979941829331861469", "-1259358775896763343051694775385987771517806" ], [ "13434870681765045829611551147217379/1026169", "67909287018595777234819162150437056031048512222738/1039509197" ], [ "-2128476924703349108908527744522501/243049", "-186847352402051756015896357808242938299356659058862/119823157" ], [ "-7364938748841807757773625709", "-1537278043534293249189664536251504660398646" ], [ "785686589410787916270883192839", "696287187804791686384919800041247659097028446" ] ], "submitter": "David Renshaw", "commentary": "Found by Elkies - Klagsbrun (2020). A curve of rank exactly 20, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:18", "updated_at": "2026-07-01 22:30:40" }, { "id": 240, "curve_key": "1392760931421184178522697:-1684829218684352826336582168404516421", "ainvs": [ "1", "-1", "0", "-29015852737941337052556", "1950033817928958623296270251397996" ], "rank_lower_bound": 19, "naive_height": 166.82945510511357, "faltings_height": 11.807447561019957, "conductor": "4408842402542882663229878258486367642610523089350038242434310458", "bad_primes": [ "2", "3", "37", "700537", "18100151", "522080015783791737432359443783998886705332553199" ], "discriminant": "-79279804082526116050199670844101862949422426192692387675453770655756", "regulator": "3358763679517923421.250400962994444783072161684480141170565001749810867504768", "points": [ [ "189487870172", "-57057674297755018" ], [ "353802982691", "-189662971158293806" ], [ "355325104871/4", "68578732568253475/8" ], [ "96203780642", "-6998703778466524" ], [ "91779544109", "7750803665126231" ], [ "-109930517796", "-61735527520602634" ], [ "1950621330926", "-2714281024184132410" ], [ "241118393681", "94720196271901040" ], [ "438116835554", "-270800205716126470" ], [ "485397239411", "-319735282350834910" ], [ "4057642908764/49", "-3680012902401856630/343" ], [ "558660770147", "-400124108256470254" ], [ "11908790775", "40077157054716614" ], [ "70449696131/4", "304048098933340693/8" ], [ "12274003974266/121", "-9457397708951999060/1331" ], [ "310042114371/4", "103281470730117533/8" ], [ "546843209204/25", "-4551468576776888702/125" ], [ "1033674110941/9", "310818670187071052/27" ], [ "7352771837441/169", "-60963589848814834888/2197" ] ], "submitter": "David Renshaw", "commentary": "Generated by rational Mestre (u,v) discriminant search: u=-3, v=-1/2, T=2229/16; exact abs-discriminant frontier triage.", "created_at": "2026-06-27 20:28:11", "updated_at": "2026-07-01 22:30:41" }, { "id": 234, "curve_key": "132982987554056065566569689:-1533719646351654702612989035132442317213", "ainvs": [ "1", "0", "1", "-2770478907376168032636869", "1775138479573443551078307813696866276" ], "rank_lower_bound": 19, "naive_height": 180.4570291062703, "faltings_height": 12.739300626770472, "conductor": "121367195540260815785780563003076660369683539545326996733038410", "bad_primes": [ "2", "3", "5", "7", "17", "7662261289645586443997884141", "4436864266498037047431696210593" ], "discriminant": "-325021758664267435353269545802736616838357964735913965003055852319441700", "regulator": "715538233792403396.6191825552383393257158767106899353176103967820819382092534", "points": [ [ "939237708948", "-39562762581743552" ], [ "1128794739009", "-293465032196013458" ], [ "921248768118", "68566127545385158" ], [ "1513488732693", "1024167210723639748" ], [ "1302979350498", "-614330875561576877" ], [ "983923487169", "-41724359928135728" ], [ "976260793788", "29802115059241003" ], [ "444611621769", "794507147074906408" ], [ "1194706365594", "-412870946793845453" ], [ "472461354918", "-756080230805621117" ], [ "957288103968", "-15852600561045077" ], [ "441115178406", "799294168677977731" ], [ "2456535768288", "-3129456458737080917" ], [ "1134305072523", "303349975608016648" ], [ "1175405086086", "377645080063855222" ], [ "982431954861", "-39342857401778228" ], [ "1401568389501", "-803340781131078404" ], [ "3567736384359/4", "933938344059977861/8" ], [ "7999856558043", "-22171701369808730492" ] ], "submitter": "RoyManami", "commentary": "Rank-19 height AND conductor record: h=180.457 (beats 190.312), log N=142.954 (beats 146.224). New 3-parameter sub-family of Mestre locus 12*p5=5*p2*p3: six roots = union of two depth-2 cubics x^3+px+q, x^3+rx+s with (p-r)(q-s)=0 (equal product), disjoint from Mestre (u,v) slice. Sextuple {0,203,204,254,370,523}, T=1303/3; base-point injection + Neron-Tate independence (19 indep).", "created_at": "2026-06-26 01:21:18", "updated_at": "2026-07-01 22:30:41" }, { "id": 160, "curve_key": "3514491230321299879293372441:-211680502645018272939774419486596394068989", "ainvs": [ "1", "-1", "0", "-73218567298360414151945259", "245000581765085750173970908953765294265" ], "rank_lower_bound": 19, "naive_height": 190.3117934033658, "faltings_height": 13.744300895897728, "conductor": "3193612294999847993147255601045320568117070630221566159701261290", "bad_primes": [ "2", "3", "5", "13", "19", "23", "29", "31", "43", "16657", "29873", "2175553654838803", "149259273131621162380144645571" ], "discriminant": "-809533846760755463057458884959863139807213987649788585257426576417408765702500", "regulator": "423708814536549837.7305076307123902558268187152316566188047402169947718710324", "points": [ [ "5549704953011", "3096083152855087682" ], [ "13056506698121", "38920366941834097517" ], [ "212900947646", "-15146680585140585313" ], [ "11070733798556", "28129323372160311527" ], [ "-4794571897764", "-22041663447857995993" ], [ "4218305221016", "-3347128863109670413" ], [ "5564955439436", "-3143585857583770243" ], [ "22114485495959/4", "-24250416298451057129/8" ], [ "5281352058011", "2370358361443582307" ], [ "6596721361454", "-7004705656918603993" ], [ "9217322631611", "-18794053512472866493" ], [ "5394735087236", "2647577719102945757" ], [ "5211535300685", "-2228287716765803110" ], [ "1852282363436", "10757982573182078807" ], [ "5751888420311", "-3761956701013851193" ], [ "5242366838660", "2287980724263201245" ], [ "-7214909794804", "19942279400410835837" ], [ "5258984909246", "-2322195160345708513" ], [ "8465892054285", "-15228327292608313255" ] ], "submitter": "cocoxhuang", "commentary": "New curve of rank ≥ 19 and and log conductor 146.224, found as the specialization t = 10806/5 of the 2-parameter Mestre rank-12 family over Q(t) used by Fermigier in 'Une courbe elliptique définie sur Q de rang ≥ 22' (Acta Arith. 82 (1997)) — here at family parameters (u,v) = (4,6), sextuple {−1146, −2304, −654, 3054, 2880, −1830} — located by a discriminant-gated exhaustive specialization scan crossed with a stagedNagao-sum sieve; 19 independent points certified by positive-definite Néron–Tate height pairing. Conductor N (log N ≈ 146.22) is squarefree apart from 3², with bad primes 2, 3², 5, 13, 19, 23, 29, 31, 43, 16657, 29873, 2175553654838803,149259273131621162380144645571.", "created_at": "2026-06-25 02:48:52", "updated_at": "2026-07-01 22:30:42" }, { "id": 90, "curve_key": "50382476556585671117196535329:-10477821508261035890454685894244764009751217", "ainvs": [ "1", "-1", "1", "-1049634928262201481608261153", "12127108227154239133795359409449990039537" ], "rank_lower_bound": 19, "naive_height": 198.26832281426988, "faltings_height": 14.48079488427077, "conductor": "1751718228241340476861580569821746434717712624403433485131514999870", "bad_primes": [ "2", "3", "5", "19", "37", "8574497", "181208356857259", "4413293403451386737", "4037539076961879275231" ], "discriminant": "10477910772405735373306877059383665173459657501610235009100067004657607156983398400", "regulator": "10371257585479461054.35598398694932580359550898384903385812428978701472381867", "points": [ [ "-8779463831497", "143755449342791785188" ], [ "24240245774135", "30446890111192735716" ], [ "-22245461900617", "156423538644219166308" ], [ "36555310855883", "-150352607095203320592" ], [ "-8080699457113", "141708301267997242020" ], [ "-14846294273337", "156326553803804543268" ], [ "36476994847767", "149582124592754048612" ], [ "-34924411096447", "78658644953248301238" ], [ "59607687170287/9", "-1996141721696772632564/27" ], [ "-26267158673455", "146883134785122482568" ], [ "-16834995329497", "-158197280846105092812" ], [ "342081548339663", "6299474080859475015468" ], [ "14189440889847", "9501613201130832932" ], [ "-20800327426857", "-157988981093285571132" ], [ "526338098596415", "12052885365530882707908" ], [ "9835031446475", "-52490357206977923004" ], [ "-37100686104889", "-1280640675347457756" ], [ "-36314897406925", "-48512158639169211804" ], [ "109946355750407/49", "33926203667374622610684/343" ] ], "submitter": "David Renshaw", "commentary": "New curve of rank >= 19 and naive height 198.268, found as the specialization T = 3251/8 (t = 3251/16) of the 2-parameter Mestre rank-12 family over Q(t) used by Fermigier in 'Une courbe elliptique definie sur Q de rang >= 22' (Acta Arith. 82 (1997), sextuple {0,55,314,378,1007,1036}), located by an exhaustive minimal-height scan crossed with a staged Nagao-sum sieve; 19 independent points certified by positive-definite Neron-Tate height pairing.", "created_at": "2026-06-11 18:51:20", "updated_at": "2026-07-01 22:30:42" }, { "id": 6, "curve_key": "99060417659838065797138894929:-28372591693416053205010168210864323950230233", "ainvs": [ "1", "-1", "1", "-2063758701246626370773726978", "32838647793306133075103747085833809114881" ], "rank_lower_bound": 19, "naive_height": 200.29658236266417, "faltings_height": 14.659918914754519, "conductor": "137377583765207564914008156149740232282282540648823380362307893110", "bad_primes": [ "2", "3", "5", "11", "17", "19", "29", "181", "73589", "2898174137", "383763976916891263992296397076513746514499" ], "discriminant": "96685523963347486558024309539021902393781074235837540884094505729945198887508377600", "regulator": "1315117024670475573.773781834878564837945549161259425446399714968525744103236", "points": [ [ "-30987785091199", "258909576181697016447" ], [ "-28889517303967", "261435145473088184895" ], [ "-71487320458034575/7921", "158780942822372481268517943/704969" ], [ "-5811612460699807/2401", "22879566180834025272238959/117649" ], [ "13824498650633", "83368523385975203271" ], [ "761787717625797737/51529", "-872181661081924408957800795/11697083" ], [ "-6768903225745", "-215633697068279328993" ], [ "-26753594302623", "-262493028566521557409" ], [ "-43702888408807", "-198899103401955431409" ], [ "-3473544815195797742755/66896041", "-720367355474239205275292698047/547142719339" ], [ "-4844597663662106466271/94965025", "67726534118909683241734294312143/925434168625" ], [ "-50527110390373", "90106963524511847991" ], [ "-45015432587775", "185797905595910416127" ], [ "-385165144189", "-183394325158424667243" ], [ "12603290965505", "93970415796904223487" ], [ "-17345672207357056815/421201", "-59881991231482114022330423257/273359449" ], [ "19374570094625", "-11265860759711920353" ], [ "-1148870915488663/169", "-474033106804620147824853/2197" ], [ "-124803550432849879/32041", "1158739203591364356910897749/5735339" ] ], "submitter": "David Renshaw", "commentary": "Found by Fermigier (1992). A historical rank ≥ 19 record, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:11:47", "updated_at": "2026-07-01 22:30:43" }, { "id": 225, "curve_key": "3596661509551646823919413851761825:-215622822992734274035136509381123478909285626560625", "ainvs": [ "1", "0", "0", "-74930448115659308831654455245038", "249563452537886885000166950293580142396024130692" ], "rank_lower_bound": 19, "naive_height": 231.79594237663446, "faltings_height": 17.053369800115544, "conductor": "305612822977927350087196447812113421419402448463094784488568526950", "bad_primes": [ "2", "3", "5", "11", "13", "17", "37", "79", "89", "6379", "28166397753511", "17930509179380017127220936003343570961" ], "discriminant": "19165526853195419552478026879498587507276138078291075987245289519588640497575159258293376000000", "regulator": "315606972157921062.9287695173038203469728749809356091846689042492693943318539", "points": [ [ "4882261436067532", "10455925384379314803634" ], [ "5710070459686252", "-88778475080120654768126" ], [ "5439993856112332", "54137910929405459176834" ], [ "4838847837782332", "16892856695505555928834" ], [ "4909045600971652", "-5312630013962056985126" ], [ "3784167745652668", "-142137498393290391417470" ], [ "4343791276627132", "-77731268535751116304766" ], [ "9639568465898602", "-650374750837561993346576" ], [ "5081960163093772", "4269076187506464793714" ], [ "1760155168566532", "-350895220399651061347166" ], [ "7231883051682652", "-293093259063868129124126" ], [ "4907638739041852", "5653244447855662364674" ], [ "20904343537780483/4", "212447116156674922899167/8" ], [ "9283304033644872", "-594973946534384517013386" ], [ "328199221517367772", "187956238503253577989464514" ], [ "4615861527249346", "-45180147467781491555720" ], [ "5156388008664772", "-17110583905139451344486" ], [ "5460212511620152", "56724109510237623938374" ], [ "-187591819551050534312/18769", "17143338594991268980322143922/2571353" ] ], "submitter": "Alexey Pozdnyakov", "commentary": "Generated with assistance from OpenAI Codex, based on GPT-5. Method: Generalized Mestre/Fermigier quartic-family search: Mestre-Nagao stage sieve, high-M rescore, exact-height triage, conductor analysis, and GP famcert certification. Candidate came from family mestre_u3_vm1 with specialization T=15619/18.", "created_at": "2026-06-25 23:18:42", "updated_at": "2026-07-01 22:31:40" }, { "id": 39, "curve_key": "-1505664758992227126399291134025916358055376960575:-260961525531561099457995721578616998276609112811677665548304369308086430625", "ainvs": [ "1", "-1", "1", "31368015812338065133318565292206590792820353345", "302038802698566087335643188429543498624522041683874493555186062568159847" ], "rank_lower_bound": 19, "naive_height": 342.70099936054265, "faltings_height": 26.616101684877602, "conductor": "565363502863493379565569563649307108299188776386222378070363275106421667133549554450", "bad_primes": [ "2", "3", "5", "7", "13", "17", "19", "47", "61", "71", "79", "83", "101", "127", "137", "229", "241", "293", "311", "523", "541", "827", "1009", "1753", "3331", "4091", "6637", "9007", "16069", "44263", "52609", "67157" ], "discriminant": "-39412228697608401237498486986030933067178933520817917493127770273615403601711081236922902433264711434650985878886431089797471702169158680576000000", "regulator": "754818738046468728836.6738481003770253068141192531193418878223715648421956502", "points": [ [ "-499155234006326082923757", "402509376386904307636088023311932246" ], [ "14139901902190764472167779", "53177370026792370019660919624681843910" ], [ "165026138566397083648829", "558309596396209485460849837556760960" ], [ "463343337635697262499219", "645016881525945325912024782026024790" ], [ "-428490311544140084738931", "458176166275251046654351925125281840" ], [ "-3076708833879670305775", "549492732729622717403028037016638716" ], [ "-37665731401560276357421", "548455893205160937812792765224762710" ], [ "-589425676522172250643765", "280659706984671022310703064556433246" ], [ "-481732375371853545253165", "418490225997586738600806619185884118" ], [ "3690858926941989311074375079", "224228684581316592254479552957274125288710" ], [ "6130049174458433978294529", "15193625860788829982804926312716459660" ], [ "99629044314032316202835", "553310835513576760020037678256777046" ], [ "-336069400109872546513581", "503527960099926164391149483607861590" ], [ "3296316187302021571885259", "6018489996489383086719697427657451870" ], [ "239450610550684059490919", "568576446830597022058144468592457990" ], [ "1116036406771517906484819", "1314196172527374653201729069765173590" ], [ "1104492511976272577147219", "1297713008801587828896903435227547990" ], [ "-556915090457934898225581", "334424588793955899397684278466715990" ], [ "142518206539016399870550419", "1701397854372553501423223969798389683990" ] ], "submitter": "David Renshaw", "commentary": "Found by Elkies (2009). A curve of rank exactly 19, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:17", "updated_at": "2026-07-01 22:31:41" }, { "id": 241, "curve_key": "144111533144482458115489:-54568282445635225440190134723996113", "ainvs": [ "1", "1", "1", "-3002323607176717877406", "63157734310826839052785319777928" ], "rank_lower_bound": 18, "naive_height": 159.97462346513558, "faltings_height": 11.139994600438078, "conductor": "441460637662701293135009928680986049143542764800100521195695", "bad_primes": [ "3", "5", "11", "419", "8761", "82207", "365239703", "34272807689", "708278198601699094328534873" ], "discriminant": "8813761630935831317440473226115886471150831299234006905672050675", "regulator": "194929077607817370.28368176934892191839151385168214171212759831564579013299", "points": [ [ "33335381221", "343488888049604" ], [ "32930342653", "-9948457849462" ], [ "58385705146", "9321742787826029" ], [ "19122392728", "-3569117322951697" ], [ "1080673626", "7740442886396429" ], [ "36080669712", "-1342491525978361" ], [ "30274713901", "-108580448018896" ], [ "238891316806", "113926312448608649" ], [ "111597974979/4", "8431760934883077/8" ], [ "29804045256", "388206891435999" ], [ "448581413029/9", "-164980061730477752/27" ], [ "68270113006", "-13280948025012256" ], [ "28159186454", "971213609060313" ], [ "2240825377969/64", "-501690777622691987/512" ], [ "755795085506", "655380411797650869" ], [ "86509789231/36", "1615773273712297909/216" ], [ "2525729586784/25", "-3515662795583333912/125" ], [ "10782499297021/400", "10774154028693723721/8000" ] ], "submitter": "David Renshaw", "commentary": "Generated by rational Mestre (u,v) discriminant search: u=-1/2, v=7/2, T=429/2; exact abs-discriminant frontier triage.", "created_at": "2026-06-27 20:28:25", "updated_at": "2026-07-01 22:31:41" }, { "id": 222, "curve_key": "159888926055372110393121:-76439655304583283704599313650092081", "ainvs": [ "1", "-1", "1", "-3331019292820252299857", "88471823270026370222046861281089" ], "rank_lower_bound": 18, "naive_height": 160.64361935761244, "faltings_height": 11.38352331175443, "conductor": "66208178266486068165130805597508803777331470494813390770", "bad_primes": [ "2", "3", "5", "37", "1601", "20646107", "548313117529", "1097006657722059559605800613523" ], "discriminant": "-1015940660555052781009392764803757490793791575413533348544972800000", "regulator": "13449739871232258.441561791518281871515000729231548101061803269736297924462", "points": [ [ "41800029847", "4719079210983476" ], [ "45876571447", "5675420192525876" ], [ "30812028847", "3884404658379476" ], [ "-12479014553", "11317972352032676" ], [ "20181196247", "5428367253417076" ], [ "32644665487", "-3810579273384364" ], [ "7360873847", "-8021938656133324" ], [ "135390399127", "-46035550207283404" ], [ "150332668567", "54637052958893876" ], [ "41235033187", "-4607646049543564" ], [ "-66023960453", "4537618321160276" ], [ "-19076706553", "12044688521663876" ], [ "56388122263", "-8940650106097612" ], [ "30824886247", "-3883638204463324" ], [ "-133558553", "9429565645552676" ], [ "-65429995013", "5129270282761136" ], [ "-62531015753", "7229109483649076" ], [ "94911552247", "-25046047487098924" ] ], "submitter": "RoyManami", "commentary": "Rank-18 naive-height record: h=160.644 (beats prior r18 height record 168.785). Found via a NEW 3-parameter sub-family of Mestre's locus 12*p5=5*p2*p3: the six roots are the union of two depth-2 cubics x^3+px+q, x^3+rx+s with (p-r)(q-s)=0 (here equal product), a slice disjoint from Mestre's (u,v) 2-parameter family. Sextuple {0,54,90,129,585,600}, T=2745/8; certified by injecting the 12 Mestre base points + Neron-Tate height-pairing independence (18 indep).", "created_at": "2026-06-25 23:05:55", "updated_at": "2026-07-01 22:31:41" }, { "id": 215, "curve_key": "2446506415688056704084825:-4479452808061048582945635774140428125", "ainvs": [ "1", "-1", "0", "-50968883660167848001767", "5184551861194511414139157441329641" ], "rank_lower_bound": 18, "naive_height": 168.78512849138983, "faltings_height": 12.056369141579205, "conductor": "5692233049957592618627388003164315988519877481788141693279603950", "bad_primes": [ "2", "3", "5", "7", "1641821", "6169133", "19823469628629700195532996237768951332198248209" ], "discriminant": "-3137843468789122931018347636744329188671582461835713108420381677437500", "regulator": "637156633527749476.44719296654871370884618117594544024639322389824847313621", "points": [ [ "124382151439", "27735090940849243" ], [ "139856705149", "-28138915164057362" ], [ "46552881319", "53969333884318678" ], [ "-208337990336", "82222089599352493" ], [ "-61208135681", "89860755353561878" ], [ "51849193319", "-51780761335278122" ], [ "138959488114", "-28021728457973057" ], [ "-159455928961", "-96215746848427157" ], [ "-54052543061", "88213549492595743" ], [ "913563564151/4", "-590980570363245601/8" ], [ "120995404208", "28087487052518733" ], [ "79772580818", "40327102621276359" ], [ "-168653863876", "94781091428093613" ], [ "-763198531", "-72273444214523922" ], [ "-981027959149/4", "-433225119414961251/8" ], [ "14219450798539", "53612964608312831743" ], [ "5762465089339/36", "7252249875034931683/216" ], [ "1114322441480", "-1154149212334190531" ] ], "submitter": "RoyManami", "commentary": "Rank-18 naive-height record: h=168.785 (beats prior r18 height record 173.853). Found via Mestre/Fermigier quartic family, sextuple mestre_ais(u=-6,v=-4), specialization T=962; certified by hyperellratpoints + Neron-Tate height-pairing independence (18 independent points).", "created_at": "2026-06-25 20:55:28", "updated_at": "2026-07-01 22:31:42" }, { "id": 147, "curve_key": "14610038047126711711563601:-56460945334792536314946406246129423705", "ainvs": [ "1", "0", "0", "-304375792648473160657575", "65348316359688219048467264095557961" ], "rank_lower_bound": 18, "naive_height": 173.85322502782768, "faltings_height": 12.356639211207849, "conductor": "21419202627854163602361890682263820949374863109651290731136698", "bad_primes": [ "2", "13", "41891672862379899773", "19665375703497198008963188787543007713701" ], "discriminant": "-40091900068111500354981590451888913785493616412246049722990279496695808", "regulator": "328231786777603894.93250258653942856450752812415634945940058946359937174453", "points": [ [ "665672967286", "397122969151581061" ], [ "528305906806", "-228032571604209659" ], [ "1621064762358", "-1957512507166916987" ], [ "141813046806", "-158227285823597147" ], [ "-449951132122", "333477709769604789" ], [ "-309393665930", "-360421436078099579" ], [ "360905322078", "50063259539241389" ], [ "-592171937290", "-194770351490693499" ], [ "-635714776838", "43943567005787149" ], [ "-169705070602", "-334835632751779451" ], [ "942176398838", "784180116547849093" ], [ "-86690933482", "-301800268160975291" ], [ "1293065099350", "1354177290109513669" ], [ "16301299883334/25", "-47452911614929710287/125" ], [ "-1515067081250/9", "9028530617604404087/27" ], [ "1623779416726", "1962768340928734021" ], [ "8394886876764/25", "-3961023236839620973/125" ], [ "2206741920622934/9025", "-63832311566160807668517/857375" ] ], "submitter": "Alexey Pozdnyakov", "commentary": "Generated with assistance from OpenAI Codex, based on GPT-5. Method: generalized Mestre/Fermigier quartic-family search, sieving Mestre (u,v) families by Mestre-Nagao score, rescoring survivors, exact-height triaging the best candidates, and certifying top score/height candidates with GP famcert. This hit came from family mestre_u2_vm4, specialization T=3433/2. GP famcert certified rank >= 18. naive-height record for rank 18: h 173.853225 < 190.774340", "created_at": "2026-06-24 03:58:36", "updated_at": "2026-07-01 22:31:42" }, { "id": 146, "curve_key": "266689216733149514371666825:-4191183778124329311075024652317330473125", "ainvs": [ "1", "0", "1", "-5556025348607281549409726", "4850907150606399626446672370423348348" ], "rank_lower_bound": 18, "naive_height": 182.544378690093, "faltings_height": 13.13781303731339, "conductor": "9120814589307131223357058040664717870768277440871516003950", "bad_primes": [ "2", "3", "5", "7", "43", "59", "109", "4289", "215507435356125127", "33984357076304424732533341601" ], "discriminant": "811199414708319103053032954716768368937184231487647543973012218268927562500", "regulator": "15766637859188658.989463949045535958988597456763312195003257509017069237721", "points": [ [ "1138306568247", "-37396477182461549" ], [ "3142973901027", "-4293677880641280689" ], [ "22836036442977", "-108566183571808263089" ], [ "5819352360516", "13022669701398455716" ], [ "1083747051522", "-320079105741957224" ], [ "587537547102", "1337666265527601361" ], [ "1630066809897", "354250759937068651" ], [ "1612633529747", "291320348627899551" ], [ "12987694770903/4", "-36696958747221394507/8" ], [ "-1913451133473", "-2911421483283008939" ], [ "4714344382032", "9134240722855240396" ], [ "720443413816947", "-19337374121787475106099" ], [ "6113245803303", "-14119082368348176203" ], [ "135656188487163/4", "-1576286240662244845417/8" ], [ "85928392099863/4", "-791921937724290050467/8" ], [ "10939779827704557/16", "-1144222024948804078791071/64" ], [ "437151064634118/169", "-6130650012875911790528/2197" ], [ "45869362770434967/3721", "9655585436888094099575191/226981" ] ], "submitter": "Alexey Pozdnyakov", "commentary": "Generated with assistance from OpenAI Codex, based on GPT-5. Method: generalized Mestre/Fermigier quartic-family search, sieving Mestre (u,v) families by Mestre-Nagao score, rescoring survivors, exact-height triaging the best candidates, and certifying top score/height candidates with GP famcert. This hit came from family mestre_u2_vm4, specialization T=826/9. GP famcert certified rank >= 18. conductor record for rank 18: log N 133.457909 < 138.988198 naive-height record for rank 18: h 182.544379 < 190.774340", "created_at": "2026-06-24 03:58:14", "updated_at": "2026-07-01 22:31:43" }, { "id": 223, "curve_key": "663709995877578012490154121:-17238768115980871498390390175127575397669", "ainvs": [ "1", "-1", "0", "-13827291580782875260211544", "19952277912018354390476665828916413300" ], "rank_lower_bound": 18, "naive_height": 185.2959588689216, "faltings_height": 13.297374883854571, "conductor": "461823608144041018516298274615425366200955693996006396490", "bad_primes": [ "2", "3", "5", "13", "17", "23", "37", "71", "83", "210790277", "21964674471223643707048498855910313431" ], "discriminant": "-2779860892166611369042972272150233658744639167770630104181815185350095802500", "regulator": "6193872058172542.0698850916042087258173155583103051269994944405161556910590", "points": [ [ "1227784678841", "2196854315198111267" ], [ "6134826146759/4", "12271431754018974331/8" ], [ "1775940598451", "998540014077880622" ], [ "2345352798959", "650718085190954954" ], [ "4720531864626", "-7737554084669710148" ], [ "2058765048281", "-459606069645467293" ], [ "2652716024666", "-1392590093046860308" ], [ "1995845950091", "552619012550347142" ], [ "4472914545539/4", "19412943032333863261/8" ], [ "-2782573437314/9", "132816334947716409844/27" ], [ "-4188336217449", "2095988457729532402" ], [ "39452428375626", "-246742606709672325398" ], [ "811017706145096/361", "3267442518430273060958/6859" ], [ "4617508820362619/2209", "-44333795825549236071734/103823" ], [ "-55963938077416/49", "2007499556130315331706/343" ], [ "2216745075624", "440102525237020394" ], [ "48407221607609/64", "-1613127844760372797391/512" ], [ "61640729874550751/27556", "-2120352425314049635526503/4574296" ] ], "submitter": "Alexey Pozdnyakov", "commentary": "Generated with assistance from OpenAI Codex, based on GPT-5. Method: Generalized Mestre/Fermigier quartic-family search: Mestre-Nagao stage sieve, high-M rescore, exact-height triage, conductor analysis, and GP famcert certification. Candidate came from family mestre_u4_vm1 with specialization T=20424/65. The same curve also appeared in the fixed sextuple_C model as T=10212/65.", "created_at": "2026-06-25 23:18:39", "updated_at": "2026-07-01 22:32:40" }, { "id": 233, "curve_key": "2258866244939094301443024001:-107387626477141067042781088591229752045249", "ainvs": [ "1", "0", "0", "-47059713436231131280063000", "124291234348538980027099166608538874432" ], "rank_lower_bound": 18, "naive_height": 188.9545271849907, "faltings_height": 13.474276471831853, "conductor": "70641518626311939848441607050541661411730955138276481056543606910", "bad_primes": [ "2", "5", "17", "37", "2021179", "81005178314881", "21895530395507809193", "3132826558949676690397" ], "discriminant": "-3639993566490641216689747626786054555892020902459466267407640882904268800000", "regulator": "536817729199543216.84665489380964481059698918907148197929007350386929801579", "points": [ [ "3239722194224", "2415430462344812008" ], [ "4499331922864", "-1907457031091338392" ], [ "3810007955424", "-547760266366600392" ], [ "3801947170944", "573657108993519768" ], [ "4147838533344", "675939614911732968" ], [ "2114906977904", "-5850124186730281752" ], [ "3404042813824", "1882105246884763368" ], [ "3609942938544", "1204989598722206568" ], [ "-7870624775696", "2668538857551716648" ], [ "30495110821168/9", "52200598495644293240/27" ], [ "-7193257748656", "-9518581173018943032" ], [ "10392518468734", "-27525637468970450962" ], [ "-2527343490552", "15069307033302071976" ], [ "32676952036364", "-182971116416499646472" ], [ "-53350585659884/9", "-376989394300296149464/27" ], [ "249957736105436/49", "-1413152923399194615756/343" ], [ "23070911278207024/5041", "782238092196844207958488/357911" ], [ "62220901060271384/7921", "10905936898001708604261312/704969" ] ], "submitter": "Alexey Pozdnyakov", "commentary": "Generated with assistance from OpenAI Codex, based on GPT-5. Method: Generalized Mestre/Fermigier quartic-family search: Mestre-Nagao stage sieve, high-M rescore, exact-height triage, and conductor analysis. Candidate came from family mestre_um2_vm4 with specialization T=2623/2.", "created_at": "2026-06-25 23:18:49", "updated_at": "2026-07-01 22:32:40" }, { "id": 98, "curve_key": "3896584043260010001837988449:-266760589578618446337644956179397294034193", "ainvs": [ "1", "-1", "1", "-81178834234583541704958093", "308750682382680533525277345167136070581" ], "rank_lower_bound": 18, "naive_height": 190.7743404293514, "faltings_height": 13.864228819343843, "conductor": "78703766491526567026991736810838956441659641637530859417300830", "bad_primes": [ "2", "5", "11", "13", "37", "89", "257", "179407", "378296549", "3677123397354060259", "260588146825380416113" ], "discriminant": "-6943255656099873846088189823650058376374717862862923480133566806025633174732800", "regulator": "73374077163928442.491606296355008103610686750381720681630217331306538695310", "points": [ [ "3690651960015", "7708299126156239672" ], [ "8639928534543", "-15884836569860852104" ], [ "1889154717743", "12733165176223454296" ], [ "5622697901945", "5483308873225802732" ], [ "3578858431597", "-8003873391211964634" ], [ "3208977934715", "9016326430132245272" ], [ "1031355338891", "-15037404035089789864" ], [ "-10461566649345", "3612321023566239032" ], [ "4283688150047", "6293763175233626296" ], [ "-9991696653105", "-11061364945580666568" ], [ "6202108581793", "6621324732236625810" ], [ "6544346735947", "7600827529120240476" ], [ "-10329673278905", "6715894460094173272" ], [ "-3048636399034585/361", "-135806158318089918059992/6859" ], [ "9517072918655", "19954232950098826312" ], [ "-76643965808353/49", "-7128301363421291146360/343" ], [ "155240191243351/25", "-830094926073353765064/125" ], [ "232411898324105/64", "4026412141608081380239/512" ] ], "submitter": "David Renshaw", "commentary": "Generated by generalized Mestre quartic-family search over (u,v) in [-50,50]^2, denominator <= 40; certified rank >= 18.", "created_at": "2026-06-23 17:40:14", "updated_at": "2026-07-01 22:32:41" }, { "id": 87, "curve_key": "6147262740489604521484573249:-476308965003899493361072044153708073135457", "ainvs": [ "1", "1", "1", "-128067973760200094197595276", "551283524310015496401062930694209419949" ], "rank_lower_bound": 18, "naive_height": 191.95741323219502, "faltings_height": 13.868689199539636, "conductor": "264272322302981023294226801215719105908812968488670879510837270", "bad_primes": [ "2", "3", "5", "7", "13", "31", "70616456303529097632401", "44220273093899778186557524292644529" ], "discriminant": "3141026088650294736579116234013850169228851134468835623113068971509123596492800", "regulator": "94146143937784834.222309731780565912552858055405979898830305053693311440478", "points": [ [ "-12733690254771", "-10832050092834520535" ], [ "7116342595149", "544176015111233065" ], [ "5775628631469", "2067176413862584585" ], [ "21771810757029", "89906083538801270665" ], [ "5328143734701", "-4492244980411323575" ], [ "7135632829749", "-874262666995103735" ], [ "5162168706573", "-5266541863119888791" ], [ "8183352551707", "7160640722003337311" ], [ "10353533303901", "18307950595024354825" ], [ "346395899469", "-22515836098862263415" ], [ "7803676630173", "5206323527832720217" ], [ "-2691447716443", "-29605322946651800879" ], [ "13890747713721", "-38112777466974057287" ], [ "22058269995549", "-91973691550298533415" ], [ "65965817063413/9", "-68106097920473546893/27" ], [ "-10499801042931", "27173753106374947945" ], [ "-12769843462147", "-10214059849860638471" ], [ "22805787632961", "-97426767137009707655" ] ], "submitter": "David Renshaw", "commentary": "New curve of rank >= 18 and naive height 191.96, found as the specialization T = 679/26 (t = 679/52) of the 2-parameter Mestre rank-12 family over Q(t) used by Fermigier in 'Une courbe elliptique definie sur Q de rang >= 22' (Acta Arith. 82 (1997), sextuple {0,55,314,378,1007,1036}), located by a staged Nagao-sum sieve over ~1.9M specializations; 18 independent points certified by positive-definite Neron-Tate height pairing.", "created_at": "2026-06-11 15:35:46", "updated_at": "2026-07-01 22:32:41" }, { "id": 161, "curve_key": "8933722659848225280802461489:-835302206420233693598092411083230386007337", "ainvs": [ "1", "-1", "1", "-186119222080171360016717948", "966784961134576267766275148260409718847" ], "rank_lower_bound": 18, "naive_height": 193.07889207019377, "faltings_height": 13.95834148292453, "conductor": "15293463740752553768990420489911754342865071901048558390", "bad_primes": [ "2", "3", "5", "13", "19", "23", "31", "37", "157", "163", "259627", "39576820317654329", "99173840155499332801" ], "discriminant": "8844410691059589836574217469287328213356255539746089974415484198645460595507200", "regulator": "1720477128584811.6188214221553592298019361587306052672826796241020503339218", "points": [ [ "-13832435464763", "-29910199873434594523" ], [ "-15238574724275", "16259276081044689533" ], [ "7127974835053", "1513322317013240093" ], [ "8923742719093", "4065823273255454213" ], [ "-15682513924187", "5350268149265678453" ], [ "2246837286493", "-23663221783492721707" ], [ "7005303109877", "2596829909785016837" ], [ "158755543668949/25", "800012588152647945433/125" ], [ "-10537945303307", "-41927054304229542907" ], [ "267554946135157/49", "3651231326520682150499/343" ], [ "86871401366965", "800240902996342804421" ], [ "-14940295804037", "-20312583017768395267" ], [ "6964686011653", "2891018295500550053" ], [ "-13822405039787", "-29975099320903870267" ], [ "9021833770315", "-4686876167691073129" ], [ "-9560836755947", "43269950494397842853" ], [ "9269843446837", "6168182658272383877" ], [ "473703694901557/49", "-2890170815217234366301/343" ] ], "submitter": "cocoxhuang", "commentary": "New curve of rank ≥ 18 and log conductor 127.067, found as the specialization t = 17142/23 of the 2-parameter Mestre rank-12 family over Q(t) used by Fermigier in 'Une courbe elliptique définie sur Q de rang ≥ 22' (Acta Arith. 82 (1997)) — here at family parameters (u,v) = (2,5), sextuple {348, −600, −216, 492, 876, −900} — located by a discriminant-gated exhaustive specialization scan crossed with a staged Nagao-sum sieve; 18 independent points certified by positive-definite Néron–Tate height pairing. Conductor N (log N ≈ 127.07) is squarefree apart from 3², with bad primes 2, 3², 5, 13, 19, 23, 31, 37, 157, 163, 259627, 39576820317654329, 99173840155499332801.", "created_at": "2026-06-25 02:54:38", "updated_at": "2026-07-01 22:32:42" }, { "id": 242, "curve_key": "14857260430275850258959924889:-1867630771212539291861082918331060446874813", "ainvs": [ "1", "0", "1", "-309526258964080213728331769", "2161609688903376164317636000643138896976" ], "rank_lower_bound": 18, "naive_height": 194.6664891321376, "faltings_height": 14.137281725412091, "conductor": "4599105032114305107130646860970904298444214155782477625288488806310", "bad_primes": [ "2", "3", "5", "17", "19", "127", "2804141", "94314161", "935018818255279", "15112931893627108554703387672603" ], "discriminant": "-120648124472674441798657772129443506216935731373216705396040170880065931985481700", "regulator": "663930766627720694.99921613424554549280042784189520445712829297273121330175", "points": [ [ "-34259847583749353581038809947044/1740884375734350625", "57715002807876692686002268734914592907285522529/2296967716638686767512359375" ], [ "7525863514713", "16075189730766472378" ], [ "-660198343256415048/37249", "-327862116505167218638695464/7189057" ], [ "441204609390461774130/27804529", "5174310853255902398806508350454/146613281417" ], [ "-29703234518830387/1681", "3168768482019896541111408/68921" ], [ "67901882238708163545/4946176", "245881666667029467951608428843/11000295424" ], [ "92226852264191646/1225", "27305643776116186645122701/42875" ], [ "10689578885677356867/1682209", "-46351324941411708197844205556/2181825073" ], [ "-27023096575177356/3721", "14402950374183416641536268/226981" ], [ "91792253098923102966/7187761", "-329018460633611310010897992958/19270387241" ], [ "437563141366809", "-9145667001043674681158" ], [ "96577445967179930765719144/4431913354521", "-708384688894540163927332586316702566707/9330112744984508931" ], [ "171621665403003489/12769", "-29899924168825630281171868/1442897" ], [ "10464469825449", "-8276152107015445988" ], [ "396411283986407896122/25270729", "-4338197160188729826225491076166/127035954683" ], [ "60092165892459/4", "-240287988841674654337/8" ], [ "11265307198908801/841", "499198982427503899130108/24389" ], [ "17714909651077851/841", "-1722454784822183933725492/24389" ] ], "submitter": "Edgar Costa", "commentary": "Rank >= 18 via 18 independent rational points with a positive-definite Neron-Tate height-pairing matrix. Mestre/Fermigier u4_v6 family, T=4694. Independently cross-verified in Sage and Magma; finite-field product independence backstop. Lower bound only; no exact-rank or upper-bound claim.", "created_at": "2026-06-29 18:45:02", "updated_at": "2026-07-01 22:32:42" }, { "id": 102, "curve_key": "17012350498427222055155079825:-2213556462276218875887383925532094329805625", "ainvs": [ "1", "-1", "1", "-354423968717233792815730830", "2561986646153119712343318111147387184797" ], "rank_lower_bound": 18, "naive_height": 195.01121127308886, "faltings_height": 14.057836969831294, "conductor": "2300421877446970281453428041661531850670855338680544890937950", "bad_primes": [ "2", "5", "11", "19", "29", "31", "67", "97", "1194256489", "31549128016279775676380684297728489659559" ], "discriminant": "13821441904429578871006127124264988888315125425652332382713954798587645696000000", "regulator": "28029078662856042.835630945014296244921316895372976514932855516252265257542", "points": [ [ "18767601803769", "-50206358006201696685" ], [ "10063361108249", "3797268895762353875" ], [ "14463240795049", "-21479423046961879325" ], [ "11845650903449", "5077316719636289875" ], [ "10180034695969", "-2987808036154910885" ], [ "27862066360089", "119650218507818587155" ], [ "11579125906089", "-3248326623828937245" ], [ "101778703261921/9", "-10313188444042479295/27" ], [ "11565547025193", "3147828265004909171" ], [ "11951310161719", "5762553083670604165" ], [ "12606449162989", "9869546987446928155" ], [ "2192939908113", "-42371005295704069429" ], [ "5454584277089", "-28125412966729382245" ], [ "29333052772769", "131926590527912797315" ], [ "471775179472089", "-10239102436034213707245" ], [ "75605547614289", "638703266730899417355" ], [ "-192585037525199/9", "503686169623613931185/27" ], [ "562483151941701/49", "848706767424556824125/343" ] ], "submitter": "Alexey Pozdnyakov", "commentary": "Generated with assistance from OpenAI Codex, based on GPT-5, during a local generalized Mestre/Fermigier quartic-family search. I searched nondegenerate Mestre (u,v) families in the box [-3,3]^2, focusing here on denominator bands 41<=n<=60 and 61<=n<=80 for specializations T=m/n. For each family I used a Mestre-Nagao sieve at prime bound M=200, rescored per-family survivors at M=1000, computed exact minimal-model heights for the global top rows, and certified the top unique models by height and score using GP famcert. This rank >= 18 conductor candidate came from family mestre_u3_vm1 at T=101/62 in the n=61..80 band; famcert certified 18 independent points, and Sage computed log conductor 138.988198, below the current ICARM rank >= 18 conductor frontier if accepted.", "created_at": "2026-06-23 23:58:09", "updated_at": "2026-07-01 22:32:42" }, { "id": 224, "curve_key": "320307201103246053866773886521:-180903144453772783632334526461612615679043181", "ainvs": [ "1", "0", "1", "-6673066689650959455557789303", "209378639414088313855940823602467379072006" ], "rank_lower_bound": 18, "naive_height": 203.81723414871612, "faltings_height": 14.78565705941443, "conductor": "572909165125068318062716813219644563458478178791019281490", "bad_primes": [ "2", "3", "5", "7", "17", "19", "23", "29", "510907", "5615780176703", "4413530848208303762736531936829" ], "discriminant": "79001812652238926310614069141452885179698753918710242185753213892099337491708562500", "regulator": "4448647520751208.6688351763983694041782506910401027616687954510865258321494", "points": [ [ "52648439943435", "63135542408283779122" ], [ "152427780121155", "-1653408591206197043978" ], [ "33382358374155", "154326358334583163522" ], [ "39388005269550", "87447073241685904477" ], [ "295951840628325", "4914890668196834574952" ], [ "50025489363555", "-27321818658715172978" ], [ "154944573237195/4", "759491082898156748051/8" ], [ "-78057807129435", "504634277034685447912" ], [ "39508640005905", "-86053348519378572728" ], [ "-71995604329495", "562699303600646341272" ], [ "89282248783980", "-570341705230530251603" ], [ "-49554040294920", "646816080171094290397" ], [ "93798472814837", "-639303212109997850622" ], [ "36963847321485", "114982967705704135642" ], [ "548034406500745/9", "-4583878846330509251656/27" ], [ "199544434564695/4", "201286633741255673801/8" ], [ "44390697022770", "25098478885753929757" ], [ "3862239457538245/144", "385208152479162663358691/1728" ] ], "submitter": "Alexey Pozdnyakov", "commentary": "Generated with assistance from OpenAI Codex, based on GPT-5. Method: Generalized Mestre/Fermigier quartic-family search: Mestre-Nagao stage sieve, high-M rescore, exact-height triage, conductor analysis, and GP famcert certification. Candidate came from family sextuple_A_m498_m216_m6_414_552_m246 with specialization T=966/145.", "created_at": "2026-06-25 23:18:40", "updated_at": "2026-07-01 22:33:40" }, { "id": 94, "curve_key": "2532074705548254150979223648161:-4022653388419864861626256258360328558188935409", "ainvs": [ "1", "0", "0", "-52751556365588628145400492670", "4655848829189654008696988351827664455270212" ], "rank_lower_bound": 18, "naive_height": 210.01977539468692, "faltings_height": 15.292901227127402, "conductor": "42682665107572966519161676237698352985292192192557605656213390", "bad_primes": [ "2", "3", "5", "11", "13", "19", "31", "43", "47", "79", "9993701", "10586665361105868350008239735615374993841941" ], "discriminant": "30329481606266546964270732939527654191747398716945306980836010472365688288267827200000", "regulator": "65246279290282792.166077869689225539534770704132049918615908028178898997037", "points": [ [ "200212896884244", "-1455981468425261901102" ], [ "128196315395964", "-10477319710847678382" ], [ "-205036993618356", "-2617647609254374848702" ], [ "105878869571004", "507455632507918359858" ], [ "144206806357572", "-218135007877172188782" ], [ "781870376871564", "-21009164439672659945982" ], [ "126484045207764", "-84483037515959599782" ], [ "-9476651858696364/49", "-947083054999256026387626/343" ], [ "18548018008764", "-1919321000484069899982" ], [ "141801409967604", "163998231164818681938" ], [ "592900127539964", "13483395378366317086418" ], [ "196304908050924", "1365719545267889883378" ], [ "19275260524764", "1909505236219745647218" ], [ "128085575458524", "22279824368584248498" ], [ "7388129939825436/49", "123606160234861151139774/343" ], [ "124886954998764", "-125293276775632768782" ], [ "127624354397784", "46988285799809697078" ], [ "1001741371622664", "-30936154910786659336782" ] ], "submitter": "David Renshaw", "commentary": "New curve of rank >= 18, found as the specialization T = 1315/12 of the 2-parameter Mestre rank-12 family over Q(t) used by Fermigier (Acta Arith. 82 (1997), sextuple {0,55,314,378,1007,1036}), located by a staged Nagao-sum sieve; submitted for its small conductor at this rank. 18 independent points certified by positive-definite Neron-Tate height pairing.", "created_at": "2026-06-12 02:09:41", "updated_at": "2026-07-01 22:33:40" }, { "id": 73, "curve_key": "6292452822641292467497471809354043276553425:-9947944978453598835810957246660421060029615177226994118242505625", "ainvs": [ "1", "0", "0", "-131092767138360259739530662694875901594863", "11513825206543517171066572416002846205241167788788151682092217" ], "rank_lower_bound": 18, "naive_height": 295.64377456906215, "faltings_height": 22.67507446318193, "conductor": "288223154151653778337307627146202434973670195623654112440970678577409850", "bad_primes": [ "2", "3", "5", "7", "11", "19", "29", "41", "43", "47", "73", "89", "131", "139", "239", "271", "353", "421", "541", "1093", "1213", "1559", "1597", "2137", "41863", "11763979", "120637667" ], "discriminant": "86914250236990244258622004202917499782878882540759655403422203387191788671749916349755506236622734346581449244863408384000000", "regulator": "79812493268609555646.384269644382536214993392374265076604666601157836610205", "points": [ [ "93922761628601831142", "172536183142846547019651383229" ], [ "-50129317073713395258", "-4237858386924835972521216331971" ], [ "-85586222512760289858", "-4701769957935207215670356805771" ], [ "-112226086720247238258", "-4981204691390384388312280789971" ], [ "-6382114264687965138", "-3514287164324372791040569133331" ], [ "-61975801758421286388", "-4404583464940048579421718953331" ], [ "8805951679868879506902", "825658977348831750165246207555549" ], [ "1063693026478103739858", "32796024716695803287365470950325" ], [ "39058033423837004262", "2540311738090645849720634292669" ], [ "306172033454233460742", "279024851751328167924122015229" ], [ "-190899016575756710598", "-5438977902059130788276281233951" ], [ "2058431396010984516618", "91997445485354914650555923853105" ], [ "-268567728817069704177/4", "-35788685082941237533491020134023/8" ], [ "350084920332270917759742", "207138045545955430308281145901442229" ], [ "233038728147301046705862", "112497177953178712068114804246204669" ], [ "523312432158494847942", "9285663592140230654731106188029" ], [ "402009359485387238802", "4876746000213639259008862651449" ], [ "5764113665374889509776198/11881", "10205816958752115947930462213486633025/1295029" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Maksym Voznyy (2023), as listed on Andrej Dujella’s high-rank elliptic-curve tables for torsion group Z/2Z. Dujella’s page lists this curve y^2 + xy = x^3 - 131092767138360259739530662694875901594863*x + 11513825206543517171066572416002846205241167788788151682092217 and the 18 independent points submitted here.", "created_at": "2026-06-01 14:26:54", "updated_at": "2026-07-01 22:33:41" }, { "id": 63, "curve_key": "82493423699285283661595499950745752858794321:-385065468974883849402843214871548112294742551077469184169304929369", "ainvs": [ "1", "0", "0", "-1718612993735110076283239582307203184558215", "445677626128337788660554947611110167094039534522619288819463817" ], "rank_lower_bound": 18, "naive_height": 303.36387744953214, "faltings_height": 23.330153180013127, "conductor": "113867782999071766513257245176182365522858404266111886760688305190", "bad_primes": [ "2", "3", "5", "7", "11", "17", "19", "23", "71", "127", "151", "179", "241", "281", "449", "503", "1223", "1361", "1399", "2017", "2083", "2293", "2843", "5651", "74923", "436549" ], "discriminant": "239065937769943972318715082774604147597436659174149372442869290187913170319391241801587631661352787169448770987206379287600000000", "regulator": "36389786154315993359.606350911054022215893750090730455834360025330154869588", "points": [ [ "-916413066082020853346", "-35369782405684521707886952377707" ], [ "195434116396250243974", "10828966775531818886070739823413" ], [ "1265034698502036374374", "17205341075829552264529317189013" ], [ "2216840056561938704974", "86776618821766613189523105183613" ], [ "1317460353165617357158", "21637731566353513111501700533141" ], [ "-419919197918370459176", "-33065253185351621643390087304037" ], [ "-29666703514651498703639/64", "-17307761259547457289873127491657319/512" ], [ "1028534464393349854573171/4", "1043092047443803329899021606077145579/8" ], [ "-4727498668121314310834/9", "-936668010323115521156024630441249/27" ], [ "2529716865202204892214", "110846353150164494211027781928133" ], [ "1199344199314248064924", "10470797020157142702308988437263" ], [ "5710166898431918122165966/9", "13644960301239082757781340351692685151/27" ], [ "-1338652300506182722676", "-18640047007082369619825194657537" ], [ "5372757156747175085983246/1369", "11786893726772068452866617397651512729/50653" ], [ "29128426914241943990086/25", "624790324363217559708520261779569/125" ], [ "4983719710548842292787296/1849", "9864984392366637148101983568300087321/79507" ], [ "10589515239193964535623854/961", "34220949959199862455549906395740174723/29791" ], [ "1023941243985123630141565744/259081", "31035258051713148261108384857840848626527/131872229" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Elkies, 2009, as listed on Andrej Dujella’s high-rank elliptic-curve tables for torsion group Z/2Z. Dujella’s rank-18 page lists this curve and the 18 independent points submitted here.", "created_at": "2026-05-29 04:07:17", "updated_at": "2026-07-01 22:33:41" }, { "id": 38, "curve_key": "1256446084449882436622961685809657787370709070536721:-44123945595377604327137242899392065760392800145803979335441014758961586480505", "ainvs": [ "1", "0", "0", "-26175960092705884096311701787701203903556438969515", "51069381476131486489742177100373772089779103253890567848326775119094885041" ], "rank_lower_bound": 18, "naive_height": 352.9803807313966, "faltings_height": 27.276890974701136, "conductor": "25725142933318819706009609339333397294766021498979988120833604560475705134314952598", "bad_primes": [ "2", "3", "7", "11", "17", "29", "31", "41", "47", "53", "79", "107", "109", "137", "149", "181", "313", "487", "2063", "3209", "3329", "6469", "10513", "15091", "18913", "23417", "32441", "47051", "131749", "643889" ], "discriminant": "21165818183141855603806249764561281249470640134499286938938173449471997510627809748094469486747147781730447578406918725587571677363134437997346291712", "regulator": "90710216574493228744.822722175775048901778182334992644131077527874357834918", "points": [ [ "3231737970618117409726910", "477734049899522560689163862493530849" ], [ "3273309887687276930053886", "677805198733135867174443509568209441" ], [ "3241000828313547371069168", "526142335107247006178138422111656119" ], [ "3255153282123996886918526", "595229162470140060497670577715082401" ], [ "26966585148660027255767774", "137677903994541133602151567271381542145" ], [ "4770216163266413128311686", "5894953964627833789313924429835166673" ], [ "2549750315144935427762366", "950643850306503428886497351231138273" ], [ "3713349702880881629511998", "2252141359101383750330981276127632993" ], [ "3542121340287734605297214", "1671112741117758577475984222816297825" ], [ "3195043262517282313487906", "228081804454697230002446209771949321" ], [ "5646805757967450691937630", "9127716726638497548695728027642334081" ], [ "14331333023247963581855678", "51180113307318927046718773937933570273" ], [ "97191985119716082916503194/25", "354969759255330273188266145971062270037/125" ], [ "1185147271535670890597935886/169", "32017173765694036331237894920457649821237/2197" ], [ "98739463791232711776641098401842/38278969", "199867994868228186007608230146203388221790785839/236831981203" ], [ "2294824481268463496752138910", "109931747845168723937902307597670194926241" ], [ "-52959420769695630752203586/9", "31331086291021542344530429579191837787/27" ], [ "154432285705268244684426598052/44521", "13312631160178214785941322691372196088408619/9393931" ] ], "submitter": "David Renshaw", "commentary": "Found by Elkies (2006). A curve of rank exactly 18, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:16", "updated_at": "2026-07-01 22:33:41" }, { "id": 159, "curve_key": "63443960979800259369:-213981041085997763351391274197", "ainvs": [ "1", "-1", "0", "-1321749187079172070", "247663242328119893241310696" ], "rank_lower_bound": 17, "naive_height": 136.79006605770067, "faltings_height": 9.455500750508849, "conductor": "84815582549450028799125905482004857770488310327228890", "bad_primes": [ "2", "5", "11", "13", "10541939988843133587887", "5626250600098076816239435529" ], "discriminant": "121286283045713541182750044839266946611798283767937312700", "regulator": "3039850234600066.55872027922921653126582444319920232392965022269415518269", "points": [ [ "17949979918713078398541/6617884875625", "-2194610111437610055414042537451414/17024674289667203125" ], [ "1069553418", "7581789774136" ], [ "1914845121457/37249", "-96411962079340334623/7189057" ], [ "212468732619099/113569", "2516334388237502223941/38272753" ], [ "2552123861358/1681", "-2875329490394494594/68921" ], [ "209259772307129/135424", "2169286183762609158459/49836032" ], [ "4700687044159/1225", "9746723920117144932/42875" ], [ "-4095591542092/3481", "-2709694728931345656/205379" ], [ "4122774202837/3721", "-2717812918567624915/226981" ], [ "-1855427204865/2401", "-3343323765033902131/117649" ], [ "-10901289782/9", "-228387183419464/27" ], [ "61860660194/25", "13765447649461718/125" ], [ "60291876902082/16129", "446623168719117198098/2048383" ], [ "13915417197647147/4583881", "1527058322512021184486745/9814089221" ], [ "2959589615", "149194293187741" ], [ "1298436411595/484", "1347253441578324783/10648" ], [ "168529858382/49", "65428320462144828/343" ] ], "submitter": "Edgar Costa", "commentary": "Rank ≥ 17. This curve is the specialization at T = 2454 of the Mestre/Fermigier family y² = r(x,T), where r is the degree-≤4 remainder in p₆(x−T)·p₆(x+T) = g(x)² − r (g monic of degree 6), with p₆(x) = ∏ᵢ(x−aᵢ) on the sextuple (a) = (−1146, −2304, −654, 3054, 2880, −1830). It was found by Alexey Pozdnyakov via a Mestre–Nagao sieve of this family for small-conductor, high-rank specializations. The rank lower bound is witnessed by the 17 listed independent rational points, whose 17×17 Néron–Tate height-pairing matrix is positive-definite; the conductor N = 2·5·11·13·10541939988843133587887·5626250600098076816239435529 is by Tate's algorithm. Proven lower bound on the Mordell–Weil rank (no exact-rank, Selmer, or BSD claim). Curve and search by Alexey Pozdnyakov.", "created_at": "2026-06-25 00:14:13", "updated_at": "2026-07-01 22:33:42" }, { "id": 232, "curve_key": "1534698822844987196425:-63817058570289535310610131073125", "ainvs": [ "1", "1", "0", "-31972892142603899925", "73862336294883442290824915625" ], "rank_lower_bound": 17, "naive_height": 146.46714664028693, "faltings_height": 10.151456350615662, "conductor": "1909993724175983557195812136598184923916762977783755050", "bad_primes": [ "2", "3", "5", "37", "58417", "1055716357", "1519100351", "3673381145443713464969597689" ], "discriminant": "-265011629229417718560918933952998158193450863167496013187500", "regulator": "16048137292672682.3031590987077593193953077608954805380268070422691776228", "points": [ [ "2664945425", "87076738827325" ], [ "-342974200", "-291183394150925" ], [ "149881895", "262818390289590" ], [ "7129560785", "-456413221223630" ], [ "-2712349675", "-375004877470550" ], [ "3147196470", "66405189566265" ], [ "4745840670", "170334561592365" ], [ "4757680200", "-171577755782925" ], [ "36110154700", "-6782709964999925" ], [ "-3062083040", "-378224349847955" ], [ "-5946513290", "-231761534756255" ], [ "3130864116", "-66704313496629" ], [ "6582355515/4", "1282598116599585/8" ], [ "29221163254", "-4908256494503999" ], [ "3272296319", "-65400846476570" ], [ "-6204206346", "-182795562231999" ], [ "6065244854", "-321036150563855" ] ], "submitter": "Alexey Pozdnyakov", "commentary": "Generated with assistance from OpenAI Codex, based on GPT-5. Method: Generalized Mestre/Fermigier quartic-family search: Mestre-Nagao stage sieve, high-M rescore, exact-height triage, and conductor analysis. Candidate came from family mestre_um2_vm6 with specialization T=1368/1.", "created_at": "2026-06-25 23:18:49", "updated_at": "2026-07-01 22:33:42" }, { "id": 229, "curve_key": "9743516533894718280825:-994784597579910445241480197858125", "ainvs": [ "1", "-1", "0", "-202989927789473297517", "1151371062064532719495377812641" ], "rank_lower_bound": 17, "naive_height": 151.96015803739886, "faltings_height": 10.582615256325571, "conductor": "30986346452249226528045018007683724606761684860484150", "bad_primes": [ "2", "3", "5", "11", "19", "12163", "26992182881", "4165307236463", "240927190974505105687" ], "discriminant": "-37375459960171559857395780326960541388085635113973452493687500", "regulator": "10448212595552444.6828230115365136877253067334401168356078834148626238197", "points": [ [ "7037990894", "-267108038594947" ], [ "10351315964", "-399127987283707" ], [ "9989176439", "-347028484970332" ], [ "8568582824", "202839608620313" ], [ "7399727885", "-233413493097169" ], [ "16003609244", "-1414777276909147" ], [ "-8335261176", "-1504734911319937" ], [ "-6402302161", "-1479370627219132" ], [ "10572774245", "-432517338548284" ], [ "-7012120786", "1493313959892293" ], [ "168235468049", "68764723550231588" ], [ "24378863795/4", "2999674420649533/8" ], [ "8862082769", "220119583035428" ], [ "213351414551/4", "95353143544380799/8" ], [ "533117324", "1021422826196063" ], [ "128876285", "-1060760459059516" ], [ "5035853549", "-506808060708412" ] ], "submitter": "Alexey Pozdnyakov", "commentary": "Generated with assistance from OpenAI Codex, based on GPT-5. Method: Generalized Mestre/Fermigier quartic-family search: Mestre-Nagao stage sieve, high-M rescore, exact-height triage, and conductor analysis. Candidate came from family mestre_u4_v6 with specialization T=2118/1.", "created_at": "2026-06-25 23:18:46", "updated_at": "2026-07-01 22:34:39" }, { "id": 230, "curve_key": "32718074950730355306057:-5967512463503514977658274315256901", "ainvs": [ "1", "-1", "0", "-681626561473549068876", "6906843129225400827472871428396" ], "rank_lower_bound": 17, "naive_height": 155.54327647319164, "faltings_height": 10.81885291959688, "conductor": "5780619339498795734844787336993702205165528840080081279506", "bad_primes": [ "2", "3", "11", "191453", "42921536569", "7087432437710059", "501282810266440502658469" ], "discriminant": "-339935100878566181983282564139251651876964088969749259722629836", "regulator": "84981055456264679.3117933149443931503433513638479714063571863841096814388", "points": [ [ "16043886086", "317326611497870" ], [ "17280696650", "536895191338436" ], [ "14167766489", "-305867682555505" ], [ "16535679509", "396241732694276" ], [ "16013767470", "313067686622764" ], [ "18926463036", "-886423879429906" ], [ "-11684436811", "-3643623591778009" ], [ "10290536150", "991084187061164" ], [ "13790371721", "-359915883876241" ], [ "12747069311", "-537914403677005" ], [ "39502758506", "-6451647565937326" ], [ "-10653514054", "3599924151318020" ], [ "13815574626", "-356041517678630" ], [ "-31989738925/4", "-27535079488466087/8" ], [ "-30174525310", "25235897491514" ], [ "2983714754780/121", "3003650211440656834/1331" ], [ "25076172373375/729", "-96749554581867301262/19683" ] ], "submitter": "Alexey Pozdnyakov", "commentary": "Generated with assistance from OpenAI Codex, based on GPT-5. Method: Generalized Mestre/Fermigier quartic-family search: Mestre-Nagao stage sieve, high-M rescore, exact-height triage, and conductor analysis. Candidate came from family mestre_u6_vm2 with specialization T=1002/1.", "created_at": "2026-06-25 23:18:47", "updated_at": "2026-07-01 22:34:40" }, { "id": 228, "curve_key": "50629669687232646175504:-11288555295116543539429879836435392", "ainvs": [ "0", "-1", "0", "-1054784785150680128656", "13065457517847594395316291335056" ], "rank_lower_bound": 17, "naive_height": 156.8364741443496, "faltings_height": 10.930791016522752, "conductor": "762280333750526640710523123060225480376357258533234759120", "bad_primes": [ "2", "3", "5", "11", "13", "2673780583403", "127364387666511563", "65221931972159992355269" ], "discriminant": "1360395974824539864077627986338200801298862217868752080515916800", "regulator": "7177491058583459.51584888066459172079556927610102951940280269290181167717", "points": [ [ "17260477840", "-40992094518804" ], [ "17246277940", "63064660477104" ], [ "24225497616", "-1315334066540780" ], [ "11406892960", "1586780838736236" ], [ "284620084", "-3572851471915440" ], [ "40638644366", "-6108611231978970" ], [ "20259905326", "107585434400730" ], [ "15878430446", "-566103220644390" ], [ "13834761766", "1058651718838230" ], [ "14208971316", "973020615775120" ], [ "-25673092112", "-4819098757440084" ], [ "21213251841", "-485848870950080" ], [ "-3324637152", "4066384915623316" ], [ "16892740438", "-260467627781334" ], [ "27534201573", "-2212999545762434" ], [ "221094522540", "-102895942434903904" ], [ "155912503597/4", "44664916210304247/8" ] ], "submitter": "Alexey Pozdnyakov", "commentary": "Generated with assistance from OpenAI Codex, based on GPT-5. Method: Generalized Mestre/Fermigier quartic-family search: Mestre-Nagao stage sieve, high-M rescore, exact-height triage, and conductor analysis. Candidate came from family mestre_u2_v5 with specialization T=4092/1.", "created_at": "2026-06-25 23:18:45", "updated_at": "2026-07-01 22:34:40" }, { "id": 231, "curve_key": "113112631658713962479617:-38911056875304869346936499385745409", "ainvs": [ "1", "0", "0", "-2356513159556540884992", "45035945457332407869731977363456" ], "rank_lower_bound": 17, "naive_height": 159.29317303495907, "faltings_height": 11.175283713126262, "conductor": "90822486465985344845462123297288403822069848710579842463634", "bad_primes": [ "2", "13", "863", "31787682271043383", "915104046386062103", "139148940743185753507" ], "discriminant": "-38688926074726301138649337130672103413740602433127643612028665856", "regulator": "28452934182224966.6012110193994401286027763716771254315599232725266211317", "points": [ [ "5248336256", "-5728242982433280" ], [ "27508940752", "1013855318237440" ], [ "23878844464", "-1543006612856576" ], [ "19737266304", "-2492731442864768" ], [ "18373533504", "-2818010805775168" ], [ "92182935072", "24721361145021440" ], [ "32273096144", "-1611878798589648" ], [ "30975335294", "-1327459810236596" ], [ "21214458350", "-2142773068047284" ], [ "64610326912", "-12747436122575360" ], [ "26832062936", "1060098553036264" ], [ "-37518610240", "-8979742835342336" ], [ "-53256582608", "4414330307269840" ], [ "-46520921028", "-7347269609014412" ], [ "1465391525200/49", "392432565204690688/343" ], [ "531387441448/9", "285394583350887880/27" ], [ "255616230159/4", "99731229782086477/8" ] ], "submitter": "Alexey Pozdnyakov", "commentary": "Generated with assistance from OpenAI Codex, based on GPT-5. Method: Generalized Mestre/Fermigier quartic-family search: Mestre-Nagao stage sieve, high-M rescore, exact-height triage, and conductor analysis. Candidate came from family mestre_um2_vm4 with specialization T=79/2.", "created_at": "2026-06-25 23:18:48", "updated_at": "2026-07-01 22:34:41" }, { "id": 227, "curve_key": "148475898081321006759169:-54029095534595783965295888107674497", "ainvs": [ "1", "1", "1", "-3093247876694187640816", "62533675385974785566766181200209" ], "rank_lower_bound": 17, "naive_height": 160.06412878638767, "faltings_height": 11.283366212963237, "conductor": "79882435688106568827071480561070837976769102565133891830", "bad_primes": [ "2", "3", "5", "13", "19", "10780355693401696197985355001494040212789352572892563" ], "discriminant": "204873677594335180956576449314454322564972911516245035826771353600", "regulator": "4198210717650867.87015792172390323128899197641083758790537897585914004614", "points": [ [ "38378058149", "-589021931997195" ], [ "164608578773", "63352818096521829" ], [ "41487998879", "2369029083476385" ], [ "39304923569", "1294228037093685" ], [ "23928280001", "-1489311095968587" ], [ "48788824931", "5268011129112945" ], [ "51542641763", "-6326948724258261" ], [ "294670874773", "-157282021941860603" ], [ "55080221653", "7698087202523749" ], [ "38320924949", "-521149441872795" ], [ "-20697282539", "-10848457303964795" ], [ "42781590599", "-2915709540138345" ], [ "45550376581", "-4018065536875755" ], [ "158788250041/9", "-99024503808654185/27" ], [ "-14288127963", "-10188882244057499" ], [ "25640408453", "279851614094949" ], [ "25664045453", "-227764996973403" ] ], "submitter": "Alexey Pozdnyakov", "commentary": "Generated with assistance from OpenAI Codex, based on GPT-5. Method: Generalized Mestre/Fermigier quartic-family search: Mestre-Nagao stage sieve, high-M rescore, exact-height triage, and conductor analysis. Candidate came from family mestre_u2_v5 with specialization T=2307/1.", "created_at": "2026-06-25 23:18:44", "updated_at": "2026-07-01 22:34:41" }, { "id": 140, "curve_key": "172514824955097031571425:-61989182116132302320876198398860625", "ainvs": [ "1", "0", "0", "-3594058853231188157738", "71746738560038807966948556541092" ], "rank_lower_bound": 17, "naive_height": 160.5143103822668, "faltings_height": 11.365072413176854, "conductor": "24091352618468926264281495356096161902461783592680683350", "bad_primes": [ "2", "3", "5", "7", "19", "113", "560341", "19999489127437", "953605040366250006249550579373" ], "discriminant": "747464088265245338817901102477124982264635888574573124981104000000", "regulator": "4037704385123208.71737976858006876166936162596647272177129510493826025144", "points": [ [ "-7569924188", "9925713157409494" ], [ "23442191212", "613551147474094" ], [ "50062657252", "4157993796611974" ], [ "22198675792", "1703655989234854" ], [ "56556047092", "7027089439562854" ], [ "18657542812", "3344418144645694" ], [ "20262264952", "2691054486995074" ], [ "45585278752", "1624039153727974" ], [ "-57042987548", "9547293955469974" ], [ "-29035006988", "12313536888779494" ], [ "-40783083188", "12267484430075494" ], [ "55535939752", "6590382158179474" ], [ "-46309488548", "11784414731841574" ], [ "21319271812", "2194043119272694" ], [ "50010148852", "4133188293153574" ], [ "45092729212", "1170578797161694" ], [ "23410910452", "661337919945574" ] ], "submitter": "David Renshaw", "commentary": null, "created_at": "2026-06-24 00:14:42", "updated_at": "2026-07-01 22:34:42" }, { "id": 96, "curve_key": "180705151207223080553017:-76912722014838801657839206874307181", "ainvs": [ "1", "0", "1", "-3764690650150480844855", "89019354183527481401653208626710" ], "rank_lower_bound": 17, "naive_height": 160.6559587349115, "faltings_height": 11.169562116753058, "conductor": "25777625822377588725485551545783271350685691069629753526342", "bad_primes": [ "2", "11", "37", "107", "12412556554267617257", "23843691413839155824600324843196647" ], "discriminant": "-8540075879702050389575912256014906231939468079986198083770051916", "regulator": "19359556026346100.6303303098718257240779383918436146546525416442360173901", "points": [ [ "90465122187", "-22109028378030402" ], [ "29636398430", "1864819852756660" ], [ "26054573840", "2935775090470274" ], [ "225363887573", "103375773706825504" ], [ "238623691977", "-113040864973820416" ], [ "17360593283", "-5375360169896206" ], [ "14618830545", "-6091655100390690" ], [ "-70533645811", "-1911027879550192" ], [ "126855903943/4", "9873344264724949/8" ], [ "164303427527/4", "-15358735695850363/8" ], [ "33812349577", "618959095388288" ], [ "37672630837", "812121700571498" ], [ "624752627246/25", "-405930157653036359/125" ], [ "33831546370", "-613808681238931" ], [ "103707900164", "-28530802496948098" ], [ "27298730108", "-2567436092943578" ], [ "11197315494189/625", "81673509203033696688/15625" ] ], "submitter": "David Renshaw", "commentary": "Generated by local generalized Mestre/Fermigier quartic-family search: Mestre parameters u=-2, v=-4, specialization T=1142. Found in uv-smoke-5 by Nagao stage-1 sieve (M=200, m<=5000, n<=12), rescored at M=1000, selected by exact minimal-height triage, and certified with famcert from quartic-side rational points.", "created_at": "2026-06-23 13:32:21", "updated_at": "2026-07-01 22:34:42" }, { "id": 97, "curve_key": "497435293115885932965361:-363056242355771001963703384937665609", "ainvs": [ "1", "0", "0", "-10363235273247623603445", "420203984207204686370404578747937" ], "rank_lower_bound": 17, "naive_height": 163.75973165737193, "faltings_height": 11.566582574023533, "conductor": "14671894523046150637866090498215403768993465347230", "bad_primes": [ "2", "5", "7", "13", "613", "1024411", "19339667", "2352864853", "5453069033", "103472067337" ], "discriminant": "-5048330766436983365467936939800781032431719923365507757411328000000", "regulator": "170247190042437.264079785742106157706972865281708359581724807739340544747", "points": [ [ "27818223314", "-12387282451487657" ], [ "259321349714", "123172612765731543" ], [ "46234992594", "-6316291369579177" ], [ "94745825874", "-16995409414228777" ], [ "60159528674", "-3805832132220377" ], [ "44236152114", "-6952523701485257" ], [ "-109704598486", "15388129882508943" ], [ "115544232594", "-27665228871893417" ], [ "41433429464", "7870784808865993" ], [ "52821878354", "-4492179472932137" ], [ "95371607394", "-17300951018924377" ], [ "57062624864", "-3828168140667857" ], [ "51138107314", "-4896893849399657" ], [ "63014091294", "-4170037882068317" ], [ "34556507314", "-10166260332899657" ], [ "42727121682", "-7444193513075305" ], [ "544874284066/9", "103516527083663981/27" ] ], "submitter": "David Renshaw", "commentary": "Generated by local generalized Mestre/Fermigier quartic-family search: Mestre parameters u=-2, v=-4, specialization T=2429/5. Found in uv-smoke-5 by Nagao stage-1 sieve (M=200, m<=5000, n<=12), rescored at M=1000, selected by exact minimal-height triage, and certified with famcert from quartic-side rational points.", "created_at": "2026-06-23 13:32:30", "updated_at": "2026-07-01 22:35:39" }, { "id": 85, "curve_key": "590444989494967182401424:-153953492867090210501562993326615232", "ainvs": [ "0", "0", "0", "-12300937281145149633363", "178186913040613669561994205239138" ], "rank_lower_bound": 17, "naive_height": 164.2054902749314, "faltings_height": 11.74463527677641, "conductor": "1482227606068422543624563099019961426673156424163925520", "bad_primes": [ "2", "3", "5", "11", "19", "29", "43", "379", "683", "77184794499853885577", "395346588657568538203" ], "discriminant": "105406467094579591296588247874235886027076144817990449651642497766400", "regulator": "1817695003829779.04293109872927351022456096418502780356708712404856098214", "points": [ [ "12769374094", "4815991895126940" ], [ "125820594673", "24946324639382916" ], [ "-103323838141", "18603782679944990" ], [ "-26083877519", "21938468531054724" ], [ "-11082842866", "17696185154606860" ], [ "13821257509", "3288271100958360" ], [ "10629645547", "6973761305127870" ], [ "105648047179", "-7603198829438850" ], [ "-109627526642", "14463170264818836" ], [ "14233545121", "2446343252265924" ], [ "157251351019", "-46177454812786110" ], [ "-22414544291", "21039141738696840" ], [ "-22992476591", "-21186336664692540" ], [ "-46119352946", "-25444106607377700" ], [ "197214452197", "73638526067350200" ], [ "3422924509", "-11667126369923640" ], [ "315386379763", "-166342028869106046" ] ], "submitter": "David Renshaw", "commentary": "New curve of rank >= 17 and naive height 164.205, found as the specialization T = 533 (i.e. t = 533/2) of the 2-parameter Mestre rank-12 family over Q(t) used by Fermigier in 'Une courbe elliptique definie sur Q de rang >= 22' (Acta Arith. 82 (1997), roots {0,55,314,378,1007,1036}), located by a staged Nagao-sum sieve over ~1.9M specializations and certified by Neron-Tate height pairing.", "created_at": "2026-06-11 15:11:12", "updated_at": "2026-07-01 22:35:40" }, { "id": 62, "curve_key": "0:785203836448278982564891253072185632", "ainvs": [ "0", "0", "0", "0", "-908800736629952526116772283648363" ], "rank_lower_bound": 17, "naive_height": 165.30250283428862, "faltings_height": 11.832864478649013, "conductor": "29733076040369196065644941545122876271020366852499870815169972871684", "bad_primes": [ "2", "3", "2195745961", "413891567044514092637683" ], "discriminant": "-356796912484430352787739298541474515252244402229998449782039674460208", "regulator": "3160234672289479672416233508.12027866509734322914490119491393668142907407", "points": [ [ "298790075644804342283980784868871/338043529350370813225", "5161349606204399690920252276261178070200995539294/6215254847769749632414964520875" ], [ "270591370675628585666599951092583/1330170417506751082249", "4204011896288889911631034546142768651699741660430/48513312301533384558486215185307" ], [ "325711162975741591923445133588371/1052929463779991398225", "-5787321283665994825424977201858275717685219519956/34166376148636124826781558558375" ], [ "1211712532726128720491413342757359/16223941796409657025", "42179303035772843722884231140233126307757275471898/65348341423942992849106321375" ], [ "1283241307160188714305985428097387/3583547576733370401961", "-45511555930423359204879858346721593921694733454080/214520974960756038450867149698709" ], [ "1193485252300183998204780009216121/12129239979490109440000", "8849454084306730092832158704556981698570573221581/1335827510135299734248783872000000" ], [ "2593675287800486498913879030036859/25807176887944349617225", "-42751818514177766312939758214885318731526695289148/4145822916881649392274547955517125" ], [ "3077300921695987166539964733710863/31522550027409402194689", "-25979911623890974715859271180333902196867407684750/5596699818568151395152805066539137" ], [ "9972014278736676784554663417348859/88394686577381806455625", "603270214553538275584451704094996369762735784335148/26280839889690220249196901117453125" ], [ "11950777486078440569963034830397097/67332468107359410846916", "1195575217084807112146406927423399551700384025930165/17471765860802684591886166103771464" ], [ "32897935731038035198457842129038763/208286420018823947157289", "5233788033009907864404702853919512596771275642180280/95058604246594295490906437252522837" ], [ "20633518747541148948800944355581657/140573422665650005479396", "2502007264930004827301247855826744471532652889922755/52705365409812184960750748589475944" ], [ "61238135983124397127054621223243257/626138814511227732809796", "-2561146939700617106205336638889703164869384364340525/495456967270561665445673889123955656" ], [ "2324524904636311085961808350770265673/9878844188981889047430144", "3418217044393581168928235873217774854469695847280879045/31049828344088581964714849238008758272" ], [ "6916737030865166929678826270687531593/20533409506285541218903104", "17973246213956629567348995135636986789359373964277484485/93044692048178135854894591156891848192" ], [ "1194537640939104093442656918171419153359/234178544464776436764868225", "41285568778238657684933397401332433223076982094874752397602/3583609287614365760517866287425122866625" ], [ "28221301625148907160620412386322712252871/1214634268989054295088514025", "4740951854035394390115339947785818419239050274616667906265794/42331951983322992425954355521097926555125" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Noam D. Elkies, Mordell curves with large rank, and Elkies, ANTS XVI, section 4.2. This is the lower-height member E_k: y^2 = x^3 + k with k = -908800736629952526116772283648363 from a 3-isogenous rank-17 pair. The submitted rational points are dual-3-isogeny images of Elkies’s 17 independent integral points on E_{-27k}.", "created_at": "2026-05-29 04:06:13", "updated_at": "2026-07-01 22:35:40" }, { "id": 219, "curve_key": "3674029243428660519943129:-4233416659961556177101757991695835933", "ainvs": [ "1", "0", "1", "-76542275904763760832149", "4899787800875052237542378473282316" ], "rank_lower_bound": 17, "naive_height": 169.68999353453933, "faltings_height": 12.182303819386327, "conductor": "7123528866816955131107217283375402978917502984525303156208430", "bad_primes": [ "2", "3", "5", "7", "19139", "4326528036023", "130255667791004904223", "3144998910355074343693" ], "discriminant": "18328723162152475758783373844977983009407970299659748341711623258000900", "regulator": "37834158862449849.9148476359393670234735188524546107191970282166892065793", "points": [ [ "-285032152137", "-59664226043498732" ], [ "-45981077007", "-91225347150091292" ], [ "-58759497417", "95887962972220708" ], [ "488170460409", "-289603777638798278" ], [ "608605557828", "428653020829183768" ], [ "-111700832262", "109799527421601418" ], [ "-9060111282", "74783195378869798" ], [ "114555611787/4", "418086210615852599/8" ], [ "38448438639", "-44874226224542888" ], [ "-108076640691", "109132194752635672" ], [ "329936981943", "124747886038034548" ], [ "-231587164254", "101021753503259227" ], [ "-277722412167", "-68823086504141492" ], [ "2073419014053", "2959722195325353418" ], [ "-3363564171036/25", "-14121414026518139497/125" ], [ "-22244479262337/121", "150338829342059702108/1331" ], [ "7726866796671/25", "12970820812301328716/125" ] ], "submitter": "Edgar Costa", "commentary": "Specialization at T = -6286/9 of the Mestre/Fermigier family y^2 = r(x,T) on the sextuple (a) = (1608, -870, -1080, -642, 1542, -558) [r = degree-<=4 remainder in p6(x-T)p6(x+T) = g^2-r, g monic degree 6, p6(x)=prod(x-a_i)]. Located by a Mestre-Nagao sieve of this family. Seventeen independent rational points were obtained by searching the low-height Mestre quartic cover and mapping to the minimal model; rank >= 17 follows from the positive-definiteness of their 17x17 Neron-Tate height-pairing matrix, verified independently in Sage and Magma. Conductor by Tate's algorithm. Proven lower bound on the rank (no exact-rank/Selmer/BSD claim).", "created_at": "2026-06-25 21:53:12", "updated_at": "2026-07-01 22:35:41" }, { "id": 216, "curve_key": "14023872430665521304793969:-52511984964278614476420631331018287753", "ainvs": [ "1", "0", "0", "-292164008972198360516541", "60777760375298123458072491954931825" ], "rank_lower_bound": 17, "naive_height": 173.70840984880002, "faltings_height": 12.17138533248092, "conductor": "8229621766982592642433268362527136439092061141374442470", "bad_primes": [ "2", "5", "11", "29", "37", "41", "61", "257", "2467", "4177", "40298971", "453713587", "575746473701180535499" ], "discriminant": "319660562013917792358156721464086339946753006672777251625449427814400", "regulator": "605706094414059953.433928438056900702744283818617694394403423434391605716", "points": [ [ "-38718845802698642844526436/105831648077401", "-375098011016010771236367933056744517569/1088737893845506994851" ], [ "370022648858", "57729896660684391" ], [ "813867360018", "-601735013298569809" ], [ "51388025011062/169", "-16041282810910889293/2197" ], [ "1742212142545418/5041", "11787551290092658020681/357911" ], [ "138421495063902/49", "-1600711642945571193647/343" ], [ "1523744473018", "1775786428781568391" ], [ "88651233252202/289", "-22108110499592240017/4913" ], [ "72435267588973433/18496", "-19318466609689318953185999/2515456" ], [ "308271320498", "2714843230760431" ], [ "82306269319150/729", "3365202373896362951845/19683" ], [ "2782325464450/9", "37096591825050565/27" ], [ "5790971164122842/18769", "6054012515438553117423/2571353" ], [ "-5029167919947/16", "-22313592130959039811/64" ], [ "1262165031167/4", "18298327651158693/8" ], [ "556113986993050/1849", "841917866240162077525/79507" ], [ "2466561849275650/289", "122260608489486622716095/4913" ] ], "submitter": "Edgar Costa", "commentary": "Specialization at T = -2407/2 of the Mestre/Fermigier family y^2 = r(x,T), where r is the degree-<=4 remainder in p6(x-T)*p6(x+T) = g(x)^2 - r (g monic of degree 6), p6(x) = prod_i(x-a_i) on the sextuple (a) = (-44, -60, -6, 110, 94, -94). Located by a Mestre-Nagao sieve of this family for small-conductor high-rank specializations. Seventeen independent rational points were found by rational-x enumeration on the quartic plus a direct minimal-model x = n/q^2 sieve; rank >= 17 follows from the positive-definiteness of their 17x17 Neron-Tate height-pairing matrix, computed independently in Sage and Magma. Conductor by Tate's algorithm. Proven lower bound on the Mordell-Weil rank (no exact-rank, Selmer, or BSD claim).", "created_at": "2026-06-25 21:52:44", "updated_at": "2026-07-01 22:35:41" }, { "id": 166, "curve_key": "176988345523030060588628721:-2354176119087252756271839420792499649481", "ainvs": [ "1", "-1", "1", "-3687257198396459595596432", "2724740878574131022947561407482810739" ], "rank_lower_bound": 17, "naive_height": 181.31437835328128, "faltings_height": 12.823862241473426, "conductor": "283351194095753440937593600911468959073927371811953815851270", "bad_primes": [ "2", "3", "5", "13", "17", "37", "612349853", "628765815428201506421019641699709465077893163" ], "discriminant": "1153068932160301343602378339411351240373132290030473551029141204930560000", "regulator": "58276555418819039.9906295005332111538296892176393442258933331924181239547", "points": [ [ "23495348691869", "-113517683577582019887" ], [ "2761907455655033/289", "-142419822038161029626487/4913" ], [ "8514920884797", "-24262997061118929199" ], [ "8105002919171097/2401", "-630759937393946987000151/117649" ], [ "57797791296753/49", "-44176752064533223057/343" ], [ "1154713200669", "-81656470446780783" ], [ "4175670052134923/3721", "-2520356457515651095617/226981" ], [ "7696384561642703/6724", "34258403517295224331743/551368" ], [ "1197083102477", "161916663098376321" ], [ "29846231699951433/24649", "727314072799652611618793/3869893" ], [ "2348554157956317/1681", "37768070556822972286521/68921" ], [ "1973500432557", "1770348455310631841" ], [ "3999708899746847", "-252954569974946715806049" ], [ "79104131395782773/51529", "9660347883013118928105483/11697083" ], [ "35416531938889181/24025", "-2613839719916380864518069/3723875" ], [ "6655658584559163/5929", "5725043071098605862733/456533" ], [ "5297203954988124077/2193361", "9133820904479001092726987241/3248367641" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥17. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=3303/2: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 17 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:00:51", "updated_at": "2026-07-01 22:35:42" }, { "id": 164, "curve_key": "236935387067387700964671009:-3663721025416020578355068837460581107377", "ainvs": [ "1", "-1", "1", "-4936153897237243770097313", "4240417853491998596755638983929308017" ], "rank_lower_bound": 17, "naive_height": 182.19859586265878, "faltings_height": 13.015054484977728, "conductor": "771012142533763749869410487682169741745976568727255733990", "bad_primes": [ "2", "3", "5", "7", "17", "31", "37", "423061", "148356075867152734210357352683148523657123407" ], "discriminant": "-70418701085637647451617054505242098864199452815445343344812504364494028800", "regulator": "11352746991815636.1172596771173279037714821445010842152357302799389226429", "points": [ [ "79842022211577577/65536", "-3137523758908713317967093/16777216" ], [ "17605289416521495703/1437601", "-72730269278505203939381021988/1723683599" ], [ "63242764747672111/42025", "-4042271758419273091936556/8615125" ], [ "26065785237334003/17161", "-1117454965119437110793952/2248091" ], [ "1659653179695", "-787111906710352892" ], [ "1942280703063", "-1407192016767310652" ], [ "2117518569559508823/1087849", "-1607545365000713659257625876/1134626507" ], [ "846101499149193063/249001", "-642052115317737566574203348/124251499" ], [ "80078043136623", "-716316208603657433972" ], [ "70814865534903", "595628366373660638308" ], [ "1128262932027343/81", "37442838388387048510652/729" ], [ "194476369296111/25", "2611815033391840449044/125" ], [ "3517375303503884887/2331729", "1696029492038458545985440364/3560550183" ], [ "10677891747607/9", "-6277332203626315604/27" ], [ "6392201212718727/5041", "-50731235352220994624260/357911" ], [ "187849311498195823/143641", "-8018354400294797441730068/54439939" ], [ "93969411133650535/68121", "-4213817666448885805296844/17779581" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥17. Mestre–Fermigier construction from the integer 6-tuple a=[240, -1692, -996, 1260, 1776, -588] with shift t=1383/4: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 17 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:00:40", "updated_at": "2026-07-01 22:35:42" }, { "id": 163, "curve_key": "1086540401236914196090781281:-35911769270128710395765526001926443074321", "ainvs": [ "1", "0", "0", "-22636258359102379085224610", "41564547766350787714309803970639244100" ], "rank_lower_bound": 17, "naive_height": 186.7637674067169, "faltings_height": 13.374448469288719, "conductor": "253719162614603331010718284712814989331353661834356557470", "bad_primes": [ "2", "3", "5", "7", "53", "113", "733", "161647205569", "98509003985351", "326103671223493957373" ], "discriminant": "-4003543597440842456820042201225551911119601184817241350745392322788249600000", "regulator": "16709644970985311.2146576317744156672737110289849683945374146824957952763", "points": [ [ "41963813756980", "-270163542135951662210" ], [ "36238334888500", "-216356341349008698290" ], [ "182911406772646514980/24433249", "-2062545615877669218157217269870/120773549807" ], [ "3761563658380", "-3104924371751178170" ], [ "98561136705675460/26569", "-12731169053892275162899910/4330747" ], [ "5138189206478380/1849", "-27554611597486805688470/79507" ], [ "66765098900596/25", "-49867050842311805146/125" ], [ "2338487488420", "-1190805577977135410" ], [ "1512239740180", "-3285026256352586210" ], [ "479603888121063940/94249", "220635347202511675210409050/28934443" ], [ "372914677744663396/9025", "226279698064107559786550834/857375" ], [ "84032515043620", "769111086316879443790" ], [ "363489079868260", "6929470060065718088590" ], [ "421895389051189638601540/64821669201", "-215147281405435544470104279938143810/16503661800243801" ], [ "9783488506214227063300/49042009", "-967427023992792791305830551022230/343441189027" ], [ "120624113779620", "-1323787532231407812210" ], [ "466217485868799160/289", "-318332482341864215809401190/4913" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥17. Mestre–Fermigier construction from the integer 6-tuple a=[-324, 24, 120, 180, 276, -276] with shift t=1355/9: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 17 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:00:14", "updated_at": "2026-07-01 22:36:39" }, { "id": 165, "curve_key": "1304157770020534929103842697:-46841312725661806886923237084796777780389", "ainvs": [ "1", "0", "1", "-27169953542094477689663390", "54214482321365567882254868382411020852" ], "rank_lower_bound": 17, "naive_height": 187.3060648689139, "faltings_height": 13.448120467938107, "conductor": "5055747593755543039707388407261319968257026830606998385654", "bad_primes": [ "2", "3", "7", "13", "23", "43", "127", "283", "78439", "212867", "29444935757", "529852873647245252778716011" ], "discriminant": "13911346942980486161686871517934679432553631814179561190438456955749228983884", "regulator": "12547921251179166.6765132651478498300270414723508308527523598212195212239", "points": [ [ "145057606823895", "1745958976309376327161" ], [ "522683745173552301/6889", "377014652656645941021061133/571787" ], [ "1583807608055806377/65536", "1950214090754745481698364267/16777216" ], [ "25355780823667965/4489", "2706170730021416266520329/300763" ], [ "356788907950971291/157609", "-129887064946403282327521283/62570773" ], [ "401598226891907958/142129", "-2799121654846465741600120/53582633" ], [ "1431224065644819/529", "8685208265999749724357/12167" ], [ "1908207155420615124/811801", "1336094215002135384439721779/731432701" ], [ "9576692532879", "-25929211779693034199" ], [ "1987724382604458/169", "-81072829838071536279911/2197" ], [ "31966715613507", "-178470005016992136695" ], [ "1219124517544603/9", "-42535846884691295629151/27" ], [ "732380383456113441843/141300769", "12185573817003989870271410080045/1679642241103" ], [ "1180741943486364", "-40572172937314419910964" ], [ "1072640820364224006741/2765569", "35127112227364754330025618978517/4599141247" ], [ "6885162389871", "13911834614960687977" ], [ "2562767237796", "-1189915762220229473" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥17. Mestre–Fermigier construction from the integer 6-tuple a=[-1146, -2304, -654, 3054, 2880, -1830] with shift t=2906/3: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 17 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:00:45", "updated_at": "2026-07-01 22:36:40" }, { "id": 189, "curve_key": "7950451352554236973757463169:-691075060315003274769578092099529022607297", "ainvs": [ "1", "0", "0", "-165634403178213270286613816", "799855393883091839227487570305634706496" ], "rank_lower_bound": 17, "naive_height": 192.72907863540027, "faltings_height": 13.967463280553337, "conductor": "8541964435847820105889019133681263942251915526797299744486693070", "bad_primes": [ "2", "5", "7", "11", "13", "31", "963617201", "156542453132674409", "182484217080500391795285792738533" ], "discriminant": "14444861515529646982364107927886725142451411941867709623555198879792460966963200", "regulator": "174183243424871196.117456918605717418856645255314545313199949862786374577", "points": [ [ "3771724805693216/25", "-230820834416887219151576/125" ], [ "220650513021776144/2209", "-102825180245912272434259304/103823" ], [ "58363265298704", "-435814179298469087992" ], [ "121206730250907905696/4490161", "-1203454835017313260439695009672/9514651159" ], [ "22736561402976", "-93742167691902387368" ], [ "89987241840089645664/5678689", "-628103463007840626180227153096/13532315887" ], [ "564525651073621876/63001", "93736751901381586320629692/15813251" ], [ "1449414454105204150/151321", "564974171284448928893290084/58863869" ], [ "273661363709014/25", "2159293870027528337902/125" ], [ "13964147866804", "34783393964917112948" ], [ "833419877491021060/56169", "533954318650346565473505592/13312053" ], [ "11612977777536", "-21035392256727489208" ], [ "3226174044596208007696/47045881", "180219935774189259169905542093208/322687697779" ], [ "573546865748084535976/11215801", "-13336036910237094494990068561432/37561717549" ], [ "43838486531548", "-278905136311491012660" ], [ "5538299763932867296/477481", "6914201219537011087158124872/329939371" ], [ "12688301446408259420456/1434818641", "-280680749199495086182087317126472/54349495302439" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥17. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=3023/1: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 17 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:02:47", "updated_at": "2026-07-01 22:36:40" }, { "id": 5, "curve_key": "886944868209120642993258774976:-835305484263775961020203080226365178089867776", "ainvs": [ "0", "1", "0", "-18478018087690013395692891145", "966788754934919721471057668405679651084743" ], "rank_lower_bound": 17, "naive_height": 206.87274282622377, "faltings_height": 14.806062691551183, "conductor": "142982366755806522511963024785618677537647778562138917247360", "bad_primes": [ "2", "3", "5", "7", "11", "13", "17", "19", "831851", "276884796725521287156064626403852388034812821" ], "discriminant": "-734320078819007341646140177726248140770988611740198639931331272725700464000000000", "regulator": "4806705005919007.18083185494710935314841121112705422150723975823444038217", "points": [ [ "987132393978079331954/12581209", "44155864801840452651790531875/44625548323" ], [ "6360438106451911/81", "832408927123396980500/729" ], [ "13270945713669554/169", "2555271033060881176965/2197" ], [ "857729078027047559/10609", "39912099554742671720961420/1092727" ], [ "79430124839906", "14615920705150940175" ], [ "4607314783851323", "-312597047325749802318252" ], [ "2573692194283109/4", "-127862522511653550016935/8" ], [ "17056161852252119/49", "-2078193005890922295589500/343" ], [ "4301027702330981171/52441", "-656366039306811393976716300/12008989" ], [ "81650469905306", "-48961061265151525875" ], [ "14515046737185390509/187489", "1323774083443035484317172500/81182737" ], [ "951024572107238604431/12243001", "528074563286919141440933947500/42838260499" ], [ "63250318746985598981/811801", "6402633724575591190797301500/731432701" ], [ "383087327491229/4", "-2199020909677353716955/8" ], [ "1738525538929581791/22201", "9310903033909158740238180/3307949" ], [ "443811832334711", "-8954505736358951228460" ], [ "4025605174011254909/27889", "-5324653812843602019420280500/4657463" ] ], "submitter": "David Renshaw", "commentary": "Found by Nagao (1992). A historical rank ≥ 17 record, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:11:46", "updated_at": "2026-07-01 22:36:40" }, { "id": 37, "curve_key": "45940295474676084120366012471924765976612825:-303780890566643118628965272922788109652779510947207621432700068125", "ainvs": [ "1", "0", "1", "-957089489055751752507625259831765957846101", "351598252970651757672333752869879740192822872602430248013582348" ], "rank_lower_bound": 17, "naive_height": 301.60774960422583, "faltings_height": 23.0392845639364, "conductor": "14736232836353426730586889330999407111621750064300242106070930802493850", "bad_primes": [ "2", "3", "5", "7", "17", "23", "37", "41", "67", "79", "97", "101", "103", "151", "229", "389", "857", "977", "983", "1499", "1531", "11719", "12907", "49639", "938573", "4153367" ], "discriminant": "2705241843717540804757209194175752043961253605237251059078988771418272961322642530579319128322487000408815513923180322592562500", "regulator": "471391564634837735.979728668453443736342367256126191005605987032331435932", "points": [ [ "1037048102780198794447", "21779881979625846052436063081576" ], [ "640151922319155456727", "1116379497785052017163204160436" ], [ "8215515531545032362283/4", "671712937153262899818205610666563/8" ], [ "-252494436924143857397/4", "162335292364600094400633862613143/8" ], [ "3033356276097950346763/4", "62943339550041852920918618404183/8" ], [ "-697039650804322492943", "26077977524885678965262644558346" ], [ "11414518769372982291382", "1215170223607965741833261815562846" ], [ "891674633090402503696", "14392420123651702644251835214211" ], [ "94919125996596744048847", "29242036615099659730101568678875176" ], [ "636565095929048595313", "542487929999796513742831489802" ], [ "4202001410067480893647", "265563310660870839720550898955176" ], [ "438987621400369262071", "4005643151170292277973367502092" ], [ "-1055345243070301537073", "13647840402839172835783080260936" ], [ "-1120995419674665243803", "3976668559063776088092945926576" ], [ "476001270991734096802", "1968115763191782636541113312386" ], [ "639949947842762264623", "1091505880400660592954070253744" ], [ "45374293429909818890233/64", "2776529523851458968917039502401287/512" ] ], "submitter": "David Renshaw", "commentary": "Found by Elkies (2005). A curve of rank exactly 17, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:15", "updated_at": "2026-07-01 22:36:41" }, { "id": 104, "curve_key": "1381248805137474000:-1643823685685922615015249600", "ainvs": [ "0", "0", "0", "-28776016773697375", "1902573710284632656267650" ], "rank_lower_bound": 16, "naive_height": 125.3336451091443, "faltings_height": 8.319114632756298, "conductor": "48425331144402124316921051447383611482427753400", "bad_primes": [ "2", "5", "90031", "533053", "5045220336795543479720942714658869" ], "discriminant": "-38740264915521699453536841157906889185942202720000", "regulator": "75232573041791.52467258926127736526099781531065272863860954044399433180", "points": [ [ "97712665", "154045371730" ], [ "99555245", "156482952370" ], [ "94394865", "165410409770" ], [ "93776785", "169506425930" ], [ "92466365", "179861364770" ], [ "89626805", "208423079030" ], [ "88694525", "219165360670" ], [ "87713925", "231006105670" ], [ "87050045", "239290996130" ], [ "85870605", "254458330030" ], [ "85428185", "260275942930" ], [ "98440675", "154237161920" ], [ "94560695", "164405897120" ], [ "78362825", "358899536830" ], [ "72920105", "438137720030" ], [ "68314345", "505541052130" ] ], "submitter": "David Renshaw", "commentary": "Found by Zev Klagsbrun.", "created_at": "2026-06-24 00:00:21", "updated_at": "2026-07-01 22:36:41" }, { "id": 91, "curve_key": "313960844280957778489:1599728574557898755218439881363", "ainvs": [ "1", "1", "0", "-6540850922519953718", "-1851537704759959591812063912" ], "rank_lower_bound": 16, "naive_height": 141.58739985684974, "faltings_height": 9.861964753778004, "conductor": "609139444635528560504687036989987088238968485253777790", "bad_primes": [ "2", "3", "5", "29", "31", "3169408342209139", "44658483011160071", "159570870149555903" ], "discriminant": "16428490821820205276811409387619951769804980047294386996300", "regulator": "876682647260930.4046834775127923693175750806292138104214702456082460330", "points": [ [ "-1579605826", "67373394280418" ], [ "-2401956187", "-1210893568960" ], [ "-3224222629/4", "-430589824752611/8" ], [ "6123597938", "-433270141862710" ], [ "-2385452686", "13313205098948" ], [ "-9579433429/4", "-70431852761111/8" ], [ "-2399183526", "5592975096618" ], [ "-297102431", "8095815100788" ], [ "-2308393444", "30768491385776" ], [ "-2312819314", "-30076910607472" ], [ "3089188419", "86154878976843" ], [ "3789601958", "-166687314398248" ], [ "-1454765926", "67713978334268" ], [ "-2193105226", "-44101458225592" ], [ "-453816781", "31990018866818" ], [ "-2028432901", "55409236846283" ] ], "submitter": "David Renshaw", "commentary": "New curve of rank >= 16 and naive height 141.587, found as the specialization T = 482 (t = 241) of the 2-parameter Mestre rank-12 family over Q(t) used by Fermigier in 'Une courbe elliptique definie sur Q de rang >= 22' (Acta Arith. 82 (1997), sextuple {0,55,314,378,1007,1036}), located by an exhaustive minimal-height scan crossed with a staged Nagao-sum sieve; 16 independent points certified by positive-definite Neron-Tate height pairing.", "created_at": "2026-06-11 18:54:07", "updated_at": "2026-07-01 22:36:42" }, { "id": 64, "curve_key": "0:-15662988285231130964521270306488", "ainvs": [ "0", "0", "1", "0", "18128458663461957134862581373" ], "rank_lower_bound": 16, "naive_height": 143.65770656998274, "faltings_height": 10.029131456623523, "conductor": "141972917837550720141347359033819641029697926227598433712323", "bad_primes": [ "3", "41", "1768630113508483622913422573" ], "discriminant": "-141972917837550720141347359033819641029697926227598433712323", "regulator": "359787206510825092.8999100746454939974782593912647860402619044739747445", "points": [ [ "1061832153", "139016765771325" ], [ "-2069581821", "96250143600728" ], [ "2323084809", "175115687339501" ], [ "2437726383", "180595326275964" ], [ "3097225419", "218722516138736" ], [ "7619958189", "678654480211793" ], [ "13493940633", "1573274288396285" ], [ "15245095569", "1887136964766261" ], [ "6376192377/4", "1191407043318535/8" ], [ "17844824169", "2387591696553138" ], [ "5948344741/9", "3664166411156266/27" ], [ "7271819311/9", "3687841575833417/27" ], [ "-11236359299/9", "3434674826973187/27" ], [ "31140274567/9", "6588845927720308/27" ], [ "-26120241831/16", "7512202525369347/64" ], [ "-1185892731/25", "16830196043015853/125" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Noam D. Elkies, \"Rank of an elliptic curve and 3-rank of a quadratic field via the Burgess bounds\", ANTS XVI. This is the minimal model (7) for E_{16D}, y^2 + y = x^3 + (D-1)/4 with D = 72513834653847828539450325493. The submitted points are the 16 independent points P_i in table (10). This is the 3-isogenous companion of the E_{-432D} model already on the board and has smaller naive height.", "created_at": "2026-05-29 04:45:01", "updated_at": "2026-07-01 22:37:39" }, { "id": 46, "curve_key": "0:422900683701240536042074298275176", "ainvs": [ "0", "0", "1", "0", "-489468383913472842641289697078" ], "rank_lower_bound": 16, "naive_height": 150.24938030199138, "faltings_height": 10.578437600957578, "conductor": "141972917837550720141347359033819641029697926227598433712323", "bad_primes": [ "3", "41", "1768630113508483622913422573" ], "discriminant": "-103498257103574474983042224735654518310649788219919258176283467", "regulator": "359787206510825092.8999100746454939974782593912647860402619044739747445", "points": [ [ "7902580710", "63670717606558" ], [ "9243066342", "547910842668385" ], [ "9384872862", "580613609811649" ], [ "10588813590", "835332795310558" ], [ "11276039694", "971735349657982" ], [ "14415958344", "1583178444925222" ], [ "14600918460", "1619646563246566" ], [ "38242987644", "7445931730687462" ], [ "31840756833/4", "977373412490165/8" ], [ "32498192145/4", "1731017073186653/8" ], [ "7916896660", "82103281566566" ], [ "8045520694", "176978571769182" ], [ "8702884360", "411934199691558" ], [ "12861701800", "1279905353076441" ], [ "22606480144", "3326205023518937" ], [ "202406423745/4", "90889676714539589/8" ] ], "submitter": "David Renshaw", "commentary": "Rank 16 Mordell curve from Noam D. Elkies, \"Rank of an elliptic curve and 3-rank of a quadratic field via the Burgess bounds\", ANTS XVI. This is the minimal model (8) for E_{-432D}, with D = 72513834653847828539450325493 = 41 * 1768630113508483622913422573: y^2 + y = x^3 - 489468383913472842641289697078. The 16 submitted points are the independent points Q_j in table (9); the paper gives conductor 27D^2 and discriminant -3^9 D^2.", "created_at": "2026-05-28 05:22:23", "updated_at": "2026-07-01 22:37:39" }, { "id": 170, "curve_key": "48116229456189285946801:-10783786994471240316075542370260201", "ainvs": [ "1", "1", "1", "-1002421447003943457225", "12481234946887000688835790191735" ], "rank_lower_bound": 16, "naive_height": 156.72670374156405, "faltings_height": 10.959174293845557, "conductor": "5859468212246645188748237684777485444087182008624730", "bad_primes": [ "2", "3", "5", "11", "13", "717496229", "2773745671", "686301029029469989491869608943" ], "discriminant": "-2831445042543812469319980404169211270210294978426890329088000000", "regulator": "755715242422743.0255065980163703580980962319872380216750682669847423293", "points": [ [ "751898661471/25", "-386048269516272384/125" ], [ "316496094584587/3249", "5364912463885590522304/185193" ], [ "21289729305823/81", "97551108696840416632/729" ], [ "373302054943", "227287226501355848" ], [ "7617130217783727/12769", "663876097852519025472376/1442897" ], [ "111751965933", "-36001073991025792" ], [ "54441827983", "-10921034534816392" ], [ "49313028731983/961", "-292136740323616231672/29791" ], [ "5650946185995727/127449", "-338053143254526723960056/45499293" ], [ "9552398586646567/499849", "-194798114168374305739256/353393243" ], [ "41878570208387/2209", "56001451768964000304/103823" ], [ "25153429663", "1783620616336328" ], [ "1427445371977/49", "970238560834856594/343" ], [ "272873933446253137/7360369", "102353737160916667605018296/19968681097" ], [ "33618451033", "-4095985604896642" ], [ "8651547690714212467/123587689", "23208991631678744562007616404/1373924338613" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=771/5: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:01:11", "updated_at": "2026-07-01 22:37:40" }, { "id": 93, "curve_key": "138640006161934808497729:-59772463447882225317598229177992033", "ainvs": [ "1", "0", "0", "-2888333461706975177036", "69181091953326695958744591394960" ], "rank_lower_bound": 16, "naive_height": 160.15170629249525, "faltings_height": 11.333679917886252, "conductor": "2134765245448937756484936235371093624988033998420670", "bad_primes": [ "2", "5", "11", "13", "19", "37", "113", "227", "7623349", "10859470707877171952191399994467277" ], "discriminant": "-525428654379535377518565414066223922787670130763896329790210867200", "regulator": "105196939230354.8297989405772618228792201639412619546898155735467441105", "points": [ [ "37416111048", "3673146956976196" ], [ "56645298168", "-9344961308712044" ], [ "31696413064", "-3078242980812300" ], [ "38166459504", "-3813140617681124" ], [ "16981380160", "-5003015355085620" ], [ "16633886728", "5073377832734196" ], [ "-4372609640", "9040303267636340" ], [ "228360089416", "-106387135973699596" ], [ "4349900728", "7529900571118196" ], [ "131565942824", "44345489428635380" ], [ "141669139928", "-50033195706517324" ], [ "25410503148", "3492047785697456" ], [ "32894041496", "3124735556097140" ], [ "-13481531176", "10279596157927796" ], [ "168559260328", "66117003092951796" ], [ "24095569528", "3684394938234996" ] ], "submitter": "David Renshaw", "commentary": "New curve of rank >= 16, found as the specialization T = 1387/2 of the 2-parameter Mestre rank-12 family over Q(t) used by Fermigier (Acta Arith. 82 (1997), sextuple {0,55,314,378,1007,1036}), located by a staged Nagao-sum sieve; submitted for its small conductor at this rank. 16 independent points certified by positive-definite Neron-Tate height pairing.", "created_at": "2026-06-12 02:09:39", "updated_at": "2026-07-01 22:37:40" }, { "id": 181, "curve_key": "191742524175704362767937:-86926153468796658699462910546782049", "ainvs": [ "1", "0", "0", "-3994635920327174224332", "100608973921885468649536289354896" ], "rank_lower_bound": 16, "naive_height": 160.90073403262164, "faltings_height": 11.328963125180563, "conductor": "2612385923234423880735212879508736416654664660133355599474", "bad_primes": [ "2", "13", "17", "31", "11959", "23251", "408088463", "1680206082665728718324081958011335561" ], "discriminant": "-293231918449934958705327201280235162673170146698422177169348591616", "regulator": "18179029818469538.42785326452085272304280545004754511587406136711446082", "points": [ [ "87115448112040/81", "811725076551771850436/729" ], [ "54853647379656152/78961", "12795840505903551940628820/22188041" ], [ "684286234212", "563721713419195768" ], [ "2077475676535/36", "1704402380888340179/216" ], [ "49957756120", "5072387202031156" ], [ "46305881816", "3863202607195380" ], [ "25675060780", "3869312490662116" ], [ "19844421339720/361", "-47088243196964595284/6859" ], [ "77021844495756/625", "-601082613854102469296/15625" ], [ "176685692356", "-70075466002658780" ], [ "391743880410598552/1771561", "-236149425639728063915026180/2357947691" ], [ "33693387294", "2065506386879344" ], [ "234291431845531154/6446521", "30331796260318243907203456/16367716819" ], [ "20977572027526184/5329", "-3037926865430113221598380/389017" ], [ "573999989640", "-432350725599294716" ], [ "27445975192", "3412733323302388" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=281/2: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:02:06", "updated_at": "2026-07-01 22:37:40" }, { "id": 167, "curve_key": "207460182766946128527249:-97821106427688674280630047678889657", "ainvs": [ "1", "-1", "1", "-4322087140978044344318", "113218873181275746647084844047757" ], "rank_lower_bound": 16, "naive_height": 161.1368968694085, "faltings_height": 11.348512992830234, "conductor": "56059062740283999533055887298056005020483418801410", "bad_primes": [ "2", "3", "5", "11", "13", "19", "23", "43", "6529", "1327973", "2253551", "11863534244867284689946519" ], "discriminant": "-370335152258882249448312295653484868140997279261599897092748492800", "regulator": "129652450970271.3285372382617348758143858676004025284640855566598266686", "points": [ [ "4932628938917", "-10954167952365331425" ], [ "446184178743003/196", "-9420897930023067416333/2744" ], [ "9544497553624449113/5909761", "-29462901095704390278748461027/14366628991" ], [ "230856486055446737/259081", "-110627478320435505484822737/131872229" ], [ "687244594339217/1369", "-17869337246424980997321/50653" ], [ "111714407682185/358801", "2273230596607997710939317/214921799" ], [ "20777055868853/841", "113138478760251712407/24389" ], [ "4297326630785/121", "2828314365172462761/1331" ], [ "38392883723", "1968003627921723" ], [ "43458840423", "2732270569874853" ], [ "205418185807217/3481", "1638702900632091787593/205379" ], [ "1009339707253977/11881", "24543070374389616153303/1295029" ], [ "50577086539596065/434281", "9870335059605680578829697/286191179" ], [ "257718003905", "126951044297053971" ], [ "686682213293", "566513798751435243" ], [ "50399148128803/36", "357407945712747949543/216" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-1146, -2304, -654, 3054, 2880, -1830] with shift t=1929/1: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:00:56", "updated_at": "2026-07-01 22:37:41" }, { "id": 179, "curve_key": "247513232241433001838817:-124318869486488173260308264018173649", "ainvs": [ "1", "1", "1", "-5156525671696520871642", "143887580422027574095483236694887" ], "rank_lower_bound": 16, "naive_height": 161.6163157236087, "faltings_height": 11.330767804286205, "conductor": "33158290302303235820773702112187496615798899795769945854", "bad_primes": [ "2", "3", "11", "13", "17", "31", "97771", "25996012748249", "27882505940647", "1034774277845385713" ], "discriminant": "-168881732015251142870523473594217561643594320461409532628190621696", "regulator": "3308054707823128.622854542820352955876197443222107463050585906051669328", "points": [ [ "1796235055657", "-2405486875415698453" ], [ "280260196900225/361", "-4672441924825390438711/6859" ], [ "33794917963577/49", "-195437004131314663299/343" ], [ "276830759494257/1369", "-4348434658144756657809/50653" ], [ "-1713074912095/169", "30688521194867715447/2197" ], [ "4976539371434433/190969", "435305469622794811181391/83453453" ], [ "150079661251/4", "14385645730349821/8" ], [ "1209129808574197/24649", "11595110519046294355607/3869893" ], [ "514060728091/9", "161327383722616421/27" ], [ "4645811742356381/27889", "291137569948406847945789/4657463" ], [ "36052656660133/81", "213816965792357845331/729" ], [ "4850146428169", "10680334290818758187" ], [ "176562964409213586197/4115607409", "336723416420365931137503833799/264028562109577" ], [ "1170267928267", "-1263655175584965025" ], [ "644082512583421135/96721", "-516876990947020027291331609/30080231" ], [ "4995701755998585/5041", "352195399117596328279933/357911" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=2577/4: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:01:56", "updated_at": "2026-07-01 22:37:41" }, { "id": 217, "curve_key": "258832074689910424162425:-131680663259199131919558611354338125", "ainvs": [ "1", "-1", "0", "-5392334889373133836717", "152408175069865597518128713267441" ], "rank_lower_bound": 16, "naive_height": 161.7313993320824, "faltings_height": 11.110527881500825, "conductor": "3865572244758359427703312649778700630767441782587550", "bad_primes": [ "2", "5", "11", "43", "47", "113233229", "2477679059415479", "12395551355742150820531" ], "discriminant": "237289500362812252947418258536952148541504393969458434158062500", "regulator": "1035675008217125.210650282138077450989277944286150483848078016626924898", "points": [ [ "-11886849988520801543020205/219020050054276", "-54739616447895393532106980635723997109/3241349121289548217976" ], [ "168878841911/4", "371469986103239/8" ], [ "93228044365251/4", "-900155943328475632151/8" ], [ "7162568606295437/169744", "4001827171142801549277/69934528" ], [ "-25775659536156/361", "-90325109045635789993/6859" ], [ "185255057136930686/4355569", "-218161975946970898534041/9090072503" ], [ "10529263292731/49", "32387293583565702279/343" ], [ "864197995016095/20449", "-66655557619693066292/2924207" ], [ "-42757882743809/529", "94403070231401230891/12167" ], [ "194866459365465611/4566769", "-861267473996490181449116/9759185353" ], [ "-8287086919/64", "6335292054507076401/512" ], [ "4898229429/16", "785813137131910117/64" ], [ "363416598", "12265747209548093" ], [ "-576940395", "12470727566797076" ], [ "700086904", "12191530118261073" ], [ "-4958351906", "13379963865786753" ] ], "submitter": "Edgar Costa", "commentary": "Specialization at T = -2068 of the Mestre/Fermigier family y^2 = r(x,T) on the sextuple (a) = (-44, -60, -6, 110, 94, -94) [r = degree-<=4 remainder in p6(x-T)p6(x+T) = g^2-r, g monic degree 6, p6(x)=prod(x-a_i)]. Located by a Mestre-Nagao sieve of this family. Sixteen independent rational points were obtained via 2-descent covers and a direct minimal-model x = n/q^2 sieve (the extra generators are high-height on the minimal model but small on the covers); rank >= 16 follows from the positive-definiteness of their 16x16 Neron-Tate height-pairing matrix, verified independently in Sage and Magma. Conductor by Tate's algorithm. Proven lower bound on the rank (no exact-rank/Selmer/BSD claim).", "created_at": "2026-06-25 21:53:06", "updated_at": "2026-07-01 22:38:38" }, { "id": 141, "curve_key": "300196990023777109586241:-166444815493197580496764574456504289", "ainvs": [ "1", "-1", "1", "-6254103958828689783047", "192644462377949614527629133622271" ], "rank_lower_bound": 16, "naive_height": 162.19994376961367, "faltings_height": 11.38894579782904, "conductor": "197022936627971949727076998839217404055886430", "bad_primes": [ "2", "3", "5", "7", "13", "4957", "662481733", "3590490075461", "2040263823050317" ], "discriminant": "-376536091399324797873843080279052128419264348845053873848320000000", "regulator": "17919212545940.76402947779221547681853496450696895958880561434176750487", "points": [ [ "53439808611", "-3322744229288306" ], [ "-21612040989", "-17824518768613106" ], [ "186751168611", "-74416602330008306" ], [ "46440583651", "-1536227171551986" ], [ "47426885331", "1646042061718414" ], [ "41564839011", "2121899058826894" ], [ "46657841601", "1553506183449034" ], [ "48467974611", "-1838227029374306" ], [ "48663260835", "1881323206446862" ], [ "-46310678589", "19569252140718094" ], [ "-84530314589", "-10830706474635506" ], [ "46331440071", "1529027925038974" ], [ "52676761011", "3060719801512894" ], [ "18056988411", "-9252130297002506" ], [ "63593508531", "-7218393089868146" ], [ "27919745211", "6308334046772494" ] ], "submitter": "Alexey Pozdnyakov", "commentary": "Generated with assistance from OpenAI Codex, based on GPT-5. Method: generalized Mestre/Fermigier quartic-family search, sieving Mestre (u,v) families by Mestre-Nagao score, rescoring survivors, exact-height triaging the best candidates, and certifying top score/height candidates with GP famcert. This hit came from the low-denominator region 1 <= n <= 40, family mestre_um2_vm4, specialization T=6951/10. GP famcert certified rank >= 16. conductor record for rank 16: log N 101.991894 < 107.496352", "created_at": "2026-06-24 01:17:09", "updated_at": "2026-07-01 22:38:38" }, { "id": 169, "curve_key": "601880103383413571773456:-466913491910981870868697429797800896", "ainvs": [ "0", "1", "0", "-12539168820487782745280", "540409134152049294268977949499028" ], "rank_lower_bound": 16, "naive_height": 164.26303564359932, "faltings_height": 11.37127205732163, "conductor": "2411607429845182224131866326159229163946583369283280", "bad_primes": [ "2", "3", "5", "11", "13", "397", "2467691", "11209488223933", "6398707603623935148158719" ], "discriminant": "16592587422778666947058928147605996735160005448846489950560000000", "regulator": "165765328977244.6178705897496116157230940079152440760508150581557650129", "points": [ [ "647952695791", "514250743863915630" ], [ "28807123943701756/67081", "4737436060388057145083250/17373979" ], [ "402262930006", "246147779511433260" ], [ "276874758276076/1521", "3885218349856161135290/59319" ], [ "6505232023429951/579121", "-8824954679706139266996990/440711081" ], [ "11163916115709526/219961", "-608011399464814507106580/103161709" ], [ "57498275851", "-3085582681209600" ], [ "750610349688556/11881", "-800744222870078099250/1295029" ], [ "64222059796", "5807235473910" ], [ "34850173328686/361", "-103930279989431903700/6859" ], [ "407905078926", "-251586046292995500" ], [ "841209759076", "-765023428267291050" ], [ "77470115910364/9", "-681812157670903246130/27" ], [ "3572373541773499774/196249", "-6751917052827679352774129700/86938307" ], [ "14946749973459694/625", "-1827323343746079478328628/15625" ], [ "74724941345461609/73984", "20306169030417705203904075/20123648" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=2892/5: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:01:06", "updated_at": "2026-07-01 22:38:39" }, { "id": 176, "curve_key": "1893955373637317153023329:-2587884603326693689092916058758460817", "ainvs": [ "1", "-1", "1", "-39457403617444107354653", "2995236809415759824391532391142437" ], "rank_lower_bound": 16, "naive_height": 167.70212899313364, "faltings_height": 11.825872743039886, "conductor": "6430563811515979376580169926167902717057516739737835010", "bad_primes": [ "2", "3", "5", "13", "17", "19", "1003771", "997188134941985254983271503747820602358073" ], "discriminant": "55901631537054437994489290286441257543068900330119374782386280857600", "regulator": "1807850498928697.709773358085301962190248467956461960084294927955712660", "points": [ [ "144809072073313", "1742582023374680171298" ], [ "119013030166267/9", "1298204132495509294076/27" ], [ "14453193394137/64", "38320836532964296481/512" ], [ "82737779887", "-17233834740637208" ], [ "1625900049957/16", "-378511156915988195/64" ], [ "99883404201318583/942841", "-1858606599514605068981832/915498611" ], [ "94373439663", "10584301043120488" ], [ "1456550150901/16", "803654378416531621/64" ], [ "118824023111096415/1456849", "31435043139868783893538264/1758416743" ], [ "664284748380235207/2036329", "-457969898199541897508533896/2905841483" ], [ "11055538290343/9", "-36305826547177738760/27" ], [ "1440208560111", "-1712732440197729176" ], [ "9113382082811474679/4644025", "-27376005199713547093247578952/10007873875" ], [ "307862534703", "-141516189959116952" ], [ "50770409072312721919/1103767729", "1310741387619272185155970466008/36670475260567" ], [ "13914484586320425535951/28771283641", "-1520159124306311998316785261631576/4880213902470061" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=2913/4: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:01:41", "updated_at": "2026-07-01 22:38:39" }, { "id": 236, "curve_key": "1928124692225363784115008:-1781647907844700462982130903918099456", "ainvs": [ "0", "0", "0", "-40169264421361745502396", "2062092485931366276599688546201504" ], "rank_lower_bound": 16, "naive_height": 167.75577030103472, "faltings_height": 12.01316038217705, "conductor": "2329523824211488655524107381802033064249929281525454944", "bad_primes": [ "2", "3", "11", "19", "31", "37", "79", "163", "1187", "2207494092079386051158169308866468641719" ], "discriminant": "2311256994351959597357327359870479823158106106819333943959339451297792", "regulator": "1575306794443644.152922817277863822046611524738222651298257620999230147", "points": [ [ "263473897774", "98835728473299268" ], [ "1694406292782", "2190584624758308540" ], [ "107788063883300184/16129", "35372203232200059004682544/2048383" ], [ "177077545644550110/1026169", "17045499838668802241704524/1039509197" ], [ "698820360270", "-561480046717032828" ], [ "6318338595346081/8100", "-486498513485524345029521/729000" ], [ "5853814809142294/34969", "35076549330988415699084/6539203" ], [ "316593635550", "-145180658240849052" ], [ "12138887762392620/63001", "607416600498386153991228/15813251" ], [ "182224971549", "-28163823213797307" ], [ "25789670918080633836/5058001", "-130868856464406362882549387748/11375444249" ], [ "63166954843761178486/375623161", "-57635319338620210678550548852/7279952483341" ], [ "431450722493976/961", "-8132585828692923037728/29791" ], [ "667137117769/4", "-11108990118817691/8" ], [ "260509642146528246/1428025", "-48423597185946377910988044/1706489875" ], [ "1244240230476587934/3545689", "-1178907764805735000176562444/6676532387" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre construction from the rational 6-tuple family a(u,v)={u,v,-u-v,-u(u+2v)²/((u-v)(2u+v)),v(2u+v)²/((u-v)(u+2v)),(u-v)²(u+v)/((u+2v)(2u+v))} at (u,v)=(-7,-4), cleared tuple [-210, -120, 330, 875, -864, -11], shift t=606/1. This family satisfies the Mestre S=0 degree-drop condition identically (two depressed cubics with equal sum of squares). Certified rank 16 via 16 independent points from a quartic point search + Néron–Tate height-pairing matrix.", "created_at": "2026-06-26 06:57:26", "updated_at": "2026-07-01 22:38:40" }, { "id": 177, "curve_key": "2881085462060791782084961:-4875875892539662193674229076352817041", "ainvs": [ "1", "0", "0", "-60022613792933162126770", "5643374875619607135973539223008900" ], "rank_lower_bound": 16, "naive_height": 168.96062805463853, "faltings_height": 11.894552101595167, "conductor": "18199850239630741806694632255103221302824470127647536470", "bad_primes": [ "2", "5", "7", "13", "14143", "61095704479", "2023834707752971", "11436658788285881147191" ], "discriminant": "81438791243922645157611866031099042317832628278473480004951449600000", "regulator": "596245719370995.9770421874122669814731596437173131542981089055244318210", "points": [ [ "5249051824831780/9", "380295550092660091636810/27" ], [ "2031681680938015/4", "91576372616636717841705/8" ], [ "2844244702180/9", "3646219393118000650/27" ], [ "290345255752", "112659987508580286" ], [ "191998494861682180/1520289", "-16484070580180227559886050/1874516337" ], [ "128168077282", "-7471078717004238" ], [ "29107455924484/225", "22258797552123760418/3375" ], [ "478620130398595780/4297329", "164229744232688785540957550/8908363017" ], [ "1771839391525/16", "1205024210105889495/64" ], [ "9629446541859620/19321", "-847256984370131499683750/2685619" ], [ "1107272954531184/2209", "-33077154155676376659774/103823" ], [ "1167606137791940/1681", "-37690495954067937504370/68921" ], [ "1128217426980", "-1172179765340581410" ], [ "115330085540", "15967736869394430" ], [ "33583372397920", "-194614399746674405950" ], [ "34375110789572", "-201537188908789628034" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=7/8: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:01:46", "updated_at": "2026-07-01 22:38:40" }, { "id": 172, "curve_key": "7004746952893110001279041:-19171511021279916944243289562775320161", "ainvs": [ "1", "-1", "1", "-145932228185273125026647", "22189248867258609149920496015950319" ], "rank_lower_bound": 16, "naive_height": 171.6929774478431, "faltings_height": 12.226738906585615, "conductor": "25711696321134843123753387143512024261228018943567970", "bad_primes": [ "2", "3", "5", "17", "19", "37", "904594283708550341", "26425921171690203209101347263" ], "discriminant": "-13801249682874560225185953107990803809947314431823775922758046720000000", "regulator": "492472590511832.3165935628802829899244997577628812687516700439862003778", "points": [ [ "77386157874918547/323761", "5719519573656468731547374/184220009" ], [ "-7462742259149/100", "180732369761751418069/1000" ], [ "72287956542747867/546121", "29080163520921100036204234/403583419" ], [ "20575728680277/121", "63706932078410060606/1331" ], [ "4080864960914165907/22363441", "4277533349985510929005804654/105756712489" ], [ "6220070339342547/28561", "131038128988252422570574/4826809" ], [ "393063052397707/1681", "2010382117608086193486/68921" ], [ "866141278428643/2209", "16497886662035545362978/103823" ], [ "293058398233203/529", "4060260064355711051762/12167" ], [ "2965346037177", "5066018568264726586" ], [ "26499730120707", "-136400935028918075714" ], [ "-1934745224044713/11881", "264246600850528066689094/1295029" ], [ "-1427985823453196933/11485321", "7628586664397936536701775734/38923752869" ], [ "-840290327883885983/19280881", "14284229246454023303849298956/84662348471" ], [ "-27104667413", "161631633543294526" ], [ "6309240191833083/29929", "145887234306918928925462/5177717" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=501/10: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:01:22", "updated_at": "2026-07-01 22:38:41" }, { "id": 180, "curve_key": "9446505198390597638003569:-27870123960027752077586111949178242697", "ainvs": [ "1", "0", "0", "-196802191633137450791741", "32257087916682386944199682931390321" ], "rank_lower_bound": 16, "naive_height": 172.52306125405872, "faltings_height": 12.305699569019374, "conductor": "118450782829775208067766915296945483495715377841216329370", "bad_primes": [ "2", "5", "11", "13", "17", "19", "23", "773", "429409", "1975931940579152131201026051304307748559" ], "discriminant": "38326897051202141276118593252174781295683024000118658651028382352998400", "regulator": "2729177115059906.931362570259210191760555095033069027438437708347161738", "points": [ [ "960455977590754/9", "29765448306343424415989/27" ], [ "1184996775243730/81", "40773516619417655092631/729" ], [ "93926389235815/36", "897876538745062245611/216" ], [ "1581689387006386/841", "61286211837482843133963/24389" ], [ "969908831278", "868211808480707547" ], [ "4642405601764/9", "7040144543520120709/27" ], [ "83217564981347554/263169", "5474064368994652222878799/135005697" ], [ "502451208525362754/1682209", "-24075177825653767340300449/2181825073" ], [ "36260210765869890/117649", "-1195938362197546351593943/40353607" ], [ "45170047429741794/134689", "-3116501815532706470874079/49430863" ], [ "396119718018", "-128278541487042673" ], [ "412971536906", "-146334814869525793" ], [ "2289306880424454322/6235009", "1517199586293175196745070511/15568817473" ], [ "969779882324404774/2712609", "-389292386971292995061885239/4467667023" ], [ "60899311327155256386/19351201", "470748801555341867901936180193/85125933199" ], [ "146178188707299250/175561", "-49106110071703841753789451/73560059" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=2594/1: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:02:01", "updated_at": "2026-07-01 22:39:38" }, { "id": 171, "curve_key": "11665590134154724806987361:-39800787938347911381207104056415738641", "ainvs": [ "1", "0", "0", "-243033127794890100145570", "46065726780475015041673462927320900" ], "rank_lower_bound": 16, "naive_height": 173.1560571786167, "faltings_height": 12.205848283826699, "conductor": "17001080899169472351595692437551130728110858780220870", "bad_primes": [ "2", "5", "17", "19", "31", "1217", "1493", "177263817121", "141808312780621", "3717407278939619" ], "discriminant": "1979571123617220930817947516260302552311906430241986844098656704000000", "regulator": "503061140161974.4973933639952823682345584297528562360756791803810313595", "points": [ [ "804935286279771142485/2640726544", "-2376316883463090509738789991615/135701655643072" ], [ "207118739029424340/436921", "-55839103538713816321049370/288804781" ], [ "13398478215906/25", "-33013515993024297636/125" ], [ "6136628085887340/10201", "-353267915888726695810770/1030301" ], [ "815682344620", "-624926083455239770" ], [ "241827032650000350/11881", "-118886100875419521952688880/1295029" ], [ "73377184147540", "-628538393730579188770" ], [ "337531713230", "49885061341813890" ], [ "965139932699684/3025", "5269925999304849389538/166375" ], [ "744546149483840/2401", "2685159731120074344270/117649" ], [ "1591318012724410480/5294601", "159323126319820860385838630/12182876901" ], [ "24442710167551460/83521", "58393952082645620285070/24137569" ], [ "2499390306144935/7396", "-31985792865062494405965/636056" ], [ "20253971560627616260/66601921", "-9123104136005428826183866290/543538277281" ], [ "2642069083118124063920/8657744209", "-14395846477111130735943458783230/805577125414823" ], [ "89533067998948/9", "-846158145396445374118/27" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-138, 138, 162, -60, -90, -12] with shift t=1355/2: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:01:16", "updated_at": "2026-07-01 22:39:38" }, { "id": 186, "curve_key": "20802047344590934454546241:-93310437331621014194286256464566516961", "ainvs": [ "1", "-1", "1", "-433375986345644467803047", "107998191356151184499232541284273119" ], "rank_lower_bound": 16, "naive_height": 174.8912809312466, "faltings_height": 12.461662233392545, "conductor": "80641417935190941169492920447777598088517349263967556694390", "bad_primes": [ "2", "3", "5", "17", "13219", "32259473509", "8349520517389", "14802985393449629740213309277" ], "discriminant": "170562405115020174321241730237321859944276566942438388072716846080000000", "regulator": "139758098201894122.2826035804257146048295377794883418723209203275293760", "points": [ [ "117823316540065899/105625", "34543457264653001082310582/34328125" ], [ "3714470393587", "7053214591294192926" ], [ "180092834680363333/21904", "76188679591183554465639687/3241792" ], [ "10159084638447", "32313984927375252466" ], [ "413500337987667", "8408397657985911069886" ], [ "8822791558227", "-26135518036715157314" ], [ "1290399889827", "-1302865125960336914" ], [ "796410792327", "-517680291105402914" ], [ "156319180011123/289", "-876615168560382378082/4913" ], [ "1577789313211767/3721", "-4942957593625439902634/226981" ], [ "4460191705964563/10609", "-11374202741656054438878/1092727" ], [ "16971071132626147/40401", "70123120568310244525166/8120601" ], [ "477541992066122787/1018081", "91420772927296554619724494/1027243729" ], [ "481359300087", "104512157193136426" ], [ "144856432632003/289", "634955008440682893518/4913" ], [ "441065116592403/529", "6949883465755085952962/12167" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=2901/10: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:02:31", "updated_at": "2026-07-01 22:39:38" }, { "id": 174, "curve_key": "107312675580214936379749161:-1108736585358091967294760632848942653909", "ainvs": [ "1", "-1", "0", "-2235680741254477841244774", "1283259936757609808257953240813254080" ], "rank_lower_bound": 16, "naive_height": 179.81336702001462, "faltings_height": 12.794622466751935, "conductor": "145828484209869499295970073108577927976542470773736770", "bad_primes": [ "2", "3", "5", "7", "11", "13", "19", "31", "89", "109", "283291781381707777230208241994647890971677" ], "discriminant": "3771450175401594216284632730407117555621648825258951862297267210498122500", "regulator": "328235901585154.0624787348667814960685903389391649690219209571650604171", "points": [ [ "1286357640157919", "46136250466930114059212" ], [ "341905934694529/9", "6317254430329285246819/27" ], [ "12049952967584", "41521230102586140032" ], [ "5541103781669", "12610637009143550369" ], [ "1247207651143055/289", "41691156313706448253285/4913" ], [ "2888647988814", "4350731104079225264" ], [ "1530043997586", "1201854479511790052" ], [ "30722339887386/25", "78231956856463548194/125" ], [ "438561162327677816/395641", "101720666381642456845702348/248858189" ], [ "944756146061", "119758896370184162" ], [ "1432445502137738039/1592644", "-11759942951095767341832359/2009916728" ], [ "370009504648442561/410881", "-4168265120033709582603338/263374721" ], [ "909974167949", "-48529226780115823" ], [ "4263622168043034/4489", "-38798376235741601969564/300763" ], [ "988130760281", "-197302981196592103" ], [ "1006411077788940806/755161", "537469664983254488916048808/656234909" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-110, 6, 60, 44, 94, -94] with shift t=2524/5: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:01:32", "updated_at": "2026-07-01 22:39:39" }, { "id": 173, "curve_key": "136036556786062058818709425:-1586606540190417935477519966720894875625", "ainvs": [ "1", "1", "1", "-2834094933042959558723113", "1836350162256247293173176506110827031" ], "rank_lower_bound": 16, "naive_height": 180.52489764411132, "faltings_height": 12.705572400524835, "conductor": "29160913962360673855189378606441749566045554275575550", "bad_primes": [ "2", "3", "5", "7", "11", "23", "37", "89", "907", "817719259139", "1216105912003", "36958698339265441" ], "discriminant": "95309530954517651197679942445741588992280745806742990246093888224000000", "regulator": "344112794184068.5132837095483207356329042306934301174297818860928244136", "points": [ [ "23133397748770775/6889", "-3141471217094483134890456/571787" ], [ "3041032595591", "-4619620262735245200" ], [ "94677769838153919/57121", "-17760474203161028032626232/13651919" ], [ "1407320880095", "-796958327588913048" ], [ "2086667697367268975/1560001", "-1290511508082994089417956712/1948441249" ], [ "1235676276895", "-470187049057551048" ], [ "317369273758475/289", "-1080809551567787511744/4913" ], [ "896313884656722515/917764", "1685321961566180587914459/879217912" ], [ "949674970078745/961", "796634291908532022882/29791" ], [ "357533678862075/361", "210056649647529768208/6859" ], [ "1024047828325", "89406907895428362" ], [ "80539355466799265/76729", "2852679029117887766832456/21253933" ], [ "67970791480505/64", "79871140889193934449/512" ], [ "1405423329526175/289", "49876486941519954749256/4913" ], [ "547498095067505304755/11377129", "-12802996373789791304733627991116/38375056117" ], [ "209690757267880795/212521", "2350255655177088892822812/97972181" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-110, 6, 60, 44, 94, -94] with shift t=3199/2: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:01:27", "updated_at": "2026-07-01 22:39:40" }, { "id": 185, "curve_key": "668466675304964315276964289:-16128439786944158266464185415356070219937", "ainvs": [ "1", "0", "0", "-13926389068853423234936756", "18667175679332355979688921260281429136" ], "rank_lower_bound": 16, "naive_height": 185.3010863316041, "faltings_height": 13.395464875988132, "conductor": "43657243228043341071443078601634748045721626333578493820310", "bad_primes": [ "2", "5", "7", "11", "23", "29", "53", "67", "127", "2749", "158647", "266621837", "124179295151", "13053706547096656897" ], "discriminant": "22324203557420732705257905143620661610548249883929708008721170067485403955200", "regulator": "836077628215322.2135091501443589696508094598206988819231525455259700725", "points": [ [ "152286804214184", "-1878728999700747104692" ], [ "765300447413032/9", "-21151245578527565284156/27" ], [ "281550663104079/4", "-4717753822362220088699/8" ], [ "486836121081064/9", "-10716771708385511474876/27" ], [ "43233200766510", "-283238750412797198806" ], [ "1184091393673305064/1071225", "2384138221407547739830063268/1108717875" ], [ "2991492522774306/2209", "157154578672959972375304/103823" ], [ "221345423961929704/133225", "16527671876532706844540028/48627125" ], [ "3240266480360533454/1934881", "-550089703964649232827177002/2691419471" ], [ "136199984927144/25", "1277794623832474021468/125" ], [ "11198548828296616/1681", "1025845985070022440673452/68921" ], [ "20410091612901109370/2337841", "84777817445557080735804895966/3574558889" ], [ "3669418539293138073664/25170289", "222205513477376556361838615217876/126279339913" ], [ "209617558513521864/124609", "958837106568577566899084/43986977" ], [ "2587523431659102184/16129", "4161116451083341379626226036/2048383" ], [ "1284015905840", "1703649433200821444" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-756, 60, 210, 486, 654, -654] with shift t=89/4: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:02:26", "updated_at": "2026-07-01 22:39:40" }, { "id": 168, "curve_key": "839985266498485472928414289:-21120087264175663299662184507972168126937", "ainvs": [ "1", "0", "0", "-17499693052051780686008631", "24444545444646300140928746272873582761" ], "rank_lower_bound": 16, "naive_height": 185.98627975097364, "faltings_height": 13.486825864689092, "conductor": "100907360654491808490244547862425873324272623451710", "bad_primes": [ "2", "3", "5", "7", "11", "13", "17", "31", "157", "821431020607251773", "2908286759476644848543" ], "discriminant": "84846485315177942827767106512600101143726929143544805106516597149399675699200", "regulator": "512847700234413.5846609390960665788755182046399204228090518568879266448", "points": [ [ "8749030777802664/2809", "-58499245138940147954151/148877" ], [ "9742749700101321/3136", "-44196140114557026187593/175616" ], [ "3123962677398", "-513183191869000203" ], [ "5367228943899126/1681", "-72934916594247767556723/68921" ], [ "5547244131698613186/1692601", "-3335257317853511364588473523/2202073901" ], [ "329295175913651664/94249", "-70598302430522311861725759/28934443" ], [ "946750469202830567910/247464361", "-14298856189125828821136719305689/3892861862891" ], [ "22196650584696645/5776", "-1639371507717248414470239/438976" ], [ "312234688643535594/69169", "-111299518938285523319516481/18191447" ], [ "1414957717465494/289", "-36808103808527122498779/4913" ], [ "9251504323176", "-25580962158180935973" ], [ "45938210681459/4", "-292653878256911864789/8" ], [ "2429484162389272848/11881", "-3786002067932057242550465007/1295029" ], [ "4031610336891147702/5329", "8094894445441753966286486445/389017" ], [ "138398804253815111544/6436369", "1599095523484338264148228166931/16329068153" ], [ "621178546558602/49", "14712769473687372255771/343" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=13506/11: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:01:01", "updated_at": "2026-07-01 22:39:41" }, { "id": 175, "curve_key": "1398063820612199224758007825:-53934088389538812494987126488734322485625", "ainvs": [ "1", "0", "0", "-29126329596087483849125163", "62423713413815642823213203923492112817" ], "rank_lower_bound": 16, "naive_height": 187.5771626848642, "faltings_height": 13.547154165453398, "conductor": "2199428785729749633481682887731454385883881335809165450", "bad_primes": [ "2", "3", "5", "11", "17", "19", "31", "43", "73", "8788437716942464397", "4825682552870583764479687" ], "discriminant": "-101999356220239409114706029907182154313690823961385846723890824782205184000000", "regulator": "393519862286889.6501035029965339047136645394249099052522877344608375237", "points": [ [ "5110339735435422", "-365320817187192230444511" ], [ "497618622929982", "-11099911453301598239391" ], [ "138299206337886", "-1625188557246978877791" ], [ "-1172961026178", "9745458965588806689" ], [ "1451827418232", "4816375036731350409" ], [ "1858357791822", "3835934257381893489" ], [ "2360920940922", "2611205774465224089" ], [ "2902124502102", "1529089488635552049" ], [ "30210396731848/9", "42492257648703299603/27" ], [ "1537958119559502/361", "27143958826181392261131/6859" ], [ "860963638869174/169", "14942816278015024293285/2197" ], [ "5129428439172", "6926964084183165339" ], [ "11766270056735352/529", "1241901508300659121973055/12167" ], [ "34194892011582", "197611265911196429409" ], [ "185624555321995496598/5257849", "2501114969242211280352058304093/12056247757" ], [ "107346062118846/25", "508148539017655964541/125" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-1146, -2304, -654, 3054, 2880, -1830] with shift t=1907/1: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:01:36", "updated_at": "2026-07-01 22:40:38" }, { "id": 188, "curve_key": "2408646883420091075699352961:-119928043257248834891063841588808215302209", "ainvs": [ "1", "0", "0", "-50180143404585230743736520", "138805605621811599445299434551151234112" ], "rank_lower_bound": 16, "naive_height": 189.17542110034245, "faltings_height": 13.645381835671836, "conductor": "3668941577753100929827306465184507435413975438465768915789448790", "bad_primes": [ "2", "3", "5", "17", "19", "307", "25219", "1733539", "159637111", "5024333263", "22802691139247", "1542477233890883" ], "discriminant": "-236561593535490259936412512778599347267021742231315444284836282901094195200000", "regulator": "136960321369372805.8656134456998812155236881189062078900842549168588408", "points": [ [ "825716257841424", "23726316375265516782648" ], [ "5525622329342214", "-410744203431915461922282" ], [ "91443162077961936/49", "-27651820816198881418299576/343" ], [ "108061543553424", "-1120974193470845681352" ], [ "56294831605250064/2401", "-12807841415343343821981768/117649" ], [ "152843981101086/49", "1218992269125288065724/343" ], [ "216195502811586/49", "622611228485546670924/343" ], [ "78123844306943544/17161", "4879495862885822492127528/2248091" ], [ "223153900113936/49", "746148161760630044424/343" ], [ "1026041124226967808/165649", "546182001320023067322019704/67419143" ], [ "783388313858485297374/111746041", "13548940718902018465047652299948/1181267399411" ], [ "39614166327566417136/4870849", "176194365587628010721239681224/10749963743" ], [ "20346720026928", "86839628682789329976" ], [ "3460337674856336/2809", "1322068898159854927638616/148877" ], [ "139553690836715272193916304/2627579402361", "1634628281522536737464060336430216707288/4259256287218536141" ], [ "70352264711064", "587207690274157016568" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=1889/6: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:02:42", "updated_at": "2026-07-01 22:40:38" }, { "id": 178, "curve_key": "11132809082008253805748920129:-1174891147887394634701529646295101307118817", "ainvs": [ "1", "-1", "1", "-231933522541838620953102503", "1359827717462320292062961439163457381087" ], "rank_lower_bound": 16, "naive_height": 193.7394988179177, "faltings_height": 13.864110287723491, "conductor": "29000334883318273334379766369449958734968994052897662690", "bad_primes": [ "2", "3", "5", "7", "11", "13", "17", "19", "23", "3294355577207735591", "1011771035177430581876701163" ], "discriminant": "-332816996738660684218455986579305397186353893376619101695289646827493750937600", "regulator": "1597599608068227.671771351581259324452130616281157992166321160531013830", "points": [ [ "39311985292419612185763/143592289", "7782675174704441356447161385367794/1720666399087" ], [ "105538367117583", "1073501176971972464158" ], [ "91326967639593", "861336775954602458668" ], [ "6271796033530083/289", "397534517028079176885710/4913" ], [ "990291502852785/64", "19668508858275895940171/512" ], [ "115917997431619887/10609", "12463248179136640291539886/1092727" ], [ "1489574132659398015195/209989081", "-25735861290825681984151049464694/3042951772771" ], [ "1805829793061414733123/213773641", "-5752256085260123832006215471362/3125584405061" ], [ "2185511165043919158579/247275625", "-2262155248862281805639449793842/3888409203125" ], [ "2476891705164386445285/274200481", "-6129616608644242348739175367258/4540485764879" ], [ "56792205463555933683/5958481", "-56477693860551678057262392322/14544652121" ], [ "187821875570497/9", "-2022024040126227255884/27" ], [ "63655414911651003319473/2863213081", "-12992980561964414368856652135329918/153207668751229" ], [ "37679089417444755/961", "-6828593700442511025350654/29791" ], [ "1147562992021579/25", "-36961131093176466599818/125" ], [ "3504789305602515987/3721", "-6560492695982148303386152730/226981" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=1851/11: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:01:51", "updated_at": "2026-07-01 22:40:38" }, { "id": 183, "curve_key": "15513705481996191519473155825:-1906469079706956357141039555142704471175625", "ainvs": [ "1", "1", "1", "-323202197541587323322357413", "2206561434845879634664264564474894007531" ], "rank_lower_bound": 16, "naive_height": 194.7345641061391, "faltings_height": 14.10580842990803, "conductor": "26383630742301813121429708006989650772225567353794271438550", "bad_primes": [ "2", "3", "5", "7", "11", "37", "191", "199", "727", "971", "2300968998537812671658704438428873760173181" ], "discriminant": "57371302902497504412701897991058097709584123040592403193410870194441105536000000", "regulator": "3263598764701149.083111521840587555373665639651653710373481275686837503", "points": [ [ "180924565545781/9", "1671651863450906696876/27" ], [ "10284908500556285015/410881", "26073080629552408556365503132/263374721" ], [ "40246669791815", "233216790682242992892" ], [ "514132005303891/4", "11549195090794268836181/8" ], [ "43617102608779015/289", "9047390518527546741124316/4913" ], [ "6630479150425740485/961", "17073238029898484032493832762/29791" ], [ "982353939816655", "-30784333888025912219828" ], [ "366922387740479", "-7020194305998509901012" ], [ "5076217626798439755/44521", "-11302507691667110491785987368/9393931" ], [ "4254910515588735/361", "-40307285259491291477612/6859" ], [ "1036183133024957815/89401", "-111930322750105636756280092/26730899" ], [ "2518861545901311335/214369", "554226264746709541557734724/99252847" ], [ "305995178478481/25", "1147752566954104940114/125" ], [ "12454361849885", "10634580957552665982" ], [ "6677202029640505315/316969", "12294351622120792432232250204/178453547" ], [ "247425027171655", "3881930663163099914172" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-110, 6, 60, 44, 94, -94] with shift t=3303/11: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:02:17", "updated_at": "2026-07-01 22:40:39" }, { "id": 190, "curve_key": "22216343063906422078063211809:-3296420228104962701344386219541803645673777", "ainvs": [ "1", "0", "0", "-462840480498050459959650246", "3815301189936260852812257361005424248676" ], "rank_lower_bound": 16, "naive_height": 195.8118771087723, "faltings_height": 14.149337069918305, "conductor": "105796999633828233559899461972111361932925512955353458023082540370", "bad_primes": [ "2", "5", "13", "17", "31", "59", "89", "157", "10687", "4882196499566433409319", "2761616040077202452279063069" ], "discriminant": "57200821486488614373193385715411199336388466289969673621065223447387681934540800", "regulator": "214216026004757241.3332241685125651774174884966523042487861989266602557", "points": [ [ "469312363122379396/2401", "-319638192435662415699200342/117649" ], [ "75846233912356206459580/680844649", "-20524335079905029521908270923524558/17765279426357" ], [ "1407409071959436/25", "-49399211055052979742974/125" ], [ "28213970481676", "-114960214643784858518" ], [ "8381669212173592671196/573842025", "-179793433897362095540192541040034/13746385708875" ], [ "3005595537603714604/227529", "273724828703003818689318386/108531333" ], [ "6603562616910707537996/497334601", "36247012241943409079347316562962/11091058936901" ], [ "8775236465006116/625", "143764105112299072042714/15625" ], [ "70619085957476", "568604587009163644042" ], [ "72904017699180", "597960596232290981546" ], [ "1201852798686484/9", "41155073865803806865014/27" ], [ "442092042524249452/1089", "293542342404627768229271914/35937" ], [ "1193902467929398508/22801", "1208195596369186292976575590/3442951" ], [ "470290282919339747660/30791401", "-3003906506723684743914116368974/170861484149" ], [ "3239968495326088/81", "-161864274616498478591270/729" ], [ "841154609703266", "-24387823893121165826928" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=1069/4: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:02:53", "updated_at": "2026-07-01 22:40:39" }, { "id": 184, "curve_key": "56120036720384132570852719825:-13306901632518891887021252432213974169545625", "ainvs": [ "1", "1", "1", "-1169167431674669428559431663", "15401506519118600679103992536133903673781" ], "rank_lower_bound": 16, "naive_height": 198.59371345224721, "faltings_height": 14.319878036317293, "conductor": "27848933853647946223887115529474676067564332131897391119550", "bad_primes": [ "2", "3", "5", "7", "11", "19", "29", "43", "73", "127", "10976906220863655801963668324973584199589113587" ], "discriminant": "-188601378345374742510097865390040902199163746431220797536602817961331170304000000", "regulator": "478580258045246.6958029338668409040983047906002491042699010978353959955", "points": [ [ "1963577490193219370285/100741369", "-4274287134927945750647758597824/1011141120653" ], [ "19725482435915", "-3767596628797241358" ], [ "44146965146747074355/2217121", "-13166106608227591906765664382/3301293169" ], [ "84992020769165771/4225", "-1304507955376817204212926/274625" ], [ "43475756338146835/1681", "-3420442022630210280753678/68921" ], [ "5169927390880715/121", "-277434016070257283551698/1331" ], [ "1045726115326939", "-33798511720116140751662" ], [ "15690799565930621520315/8473921", "1965142616992608074707764709942702/24667584031" ], [ "295490081647955", "5046826565144989978122" ], [ "1957220904751499/25", "79422926794702239882258/125" ], [ "134051813545315/9", "-970332229292678234186/27" ], [ "23850116890041395/1369", "-906698356527892497902934/50653" ], [ "17622325019835", "-16450132122712468718" ], [ "18011746641515", "-13644940173569852958" ], [ "2649576959335347595/88209", "2251205537386863546278362306/26198073" ], [ "38852091226715", "-169184993480190893358" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-1146, -2304, -654, 3054, 2880, -1830] with shift t=2919/8: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:02:21", "updated_at": "2026-07-01 22:40:40" }, { "id": 187, "curve_key": "57086385615639196900478739409:-13643169732991798117322623791734476775741977", "ainvs": [ "1", "0", "0", "-1189299700325816602093307071", "15790705709481148712666898533494358863001" ], "rank_lower_bound": 16, "naive_height": 198.6436258325694, "faltings_height": 14.28113104372204, "conductor": "6478350825852857868575711796470819320010194843359558272176170", "bad_primes": [ "2", "3", "5", "7", "13", "23", "31", "43", "3277612446733284530993", "1816531268684434524436436201159" ], "discriminant": "-57756388763308488291464979237750795586612877872758868116877248909627520480051200", "regulator": "6104323411601247.835938464639324428147234046756706737038186530028901018", "points": [ [ "697576442233977388406/177241", "-18423455744863744566579326547943/74618461" ], [ "1617980147791814", "-65067171076314760848307" ], [ "8996486721959734/9", "-852813860149243222627361/27" ], [ "2089299406246950440/18769", "-2889502216708937627799054803/2571353" ], [ "4959252092193526/361", "310190738443800799859023/6859" ], [ "810579905488502486/52441", "398415122436772395438033593/12008989" ], [ "27510586103866689230/1530169", "28071482427182809558959004193/1892819053" ], [ "18427107540326", "11507788246080029357" ], [ "3346688353809670/169", "4876157311998696125137/2197" ], [ "184998133634752/9", "146224271703375470585/27" ], [ "23177380051814", "26010241714416410093" ], [ "39043352706710", "169922010875689734941" ], [ "59337157252816", "392608088389820811787" ], [ "30436825206632108486/398161", "153149547828250401696053592347/251239591" ], [ "17545171071416", "18033907841991859187" ], [ "19600296319046", "3155410596522222317" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=842/9: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:02:37", "updated_at": "2026-07-01 22:40:40" }, { "id": 182, "curve_key": "33868439134455088165022684851681:-197051210250681199142764101862645154541055843921", "ainvs": [ "1", "0", "0", "-705592481967814336771305934410", "228068530382732810578825694652792716443320100" ], "rank_lower_bound": 16, "naive_height": 217.80010911472627, "faltings_height": 15.876700155877355, "conductor": "15495342387615096338864350597570025083525109638868085907030", "bad_primes": [ "2", "5", "7", "13", "17", "19", "23", "31", "157", "2520531720283", "36758332594990127793731213603179" ], "discriminant": "11765406596767129399549312051539530956680847347822041839788385072806810895769600000000000", "regulator": "398172387122838.1436246285395286883159166095411598056198651333868740662", "points": [ [ "85806452822380243564820/151321", "25135032253556314427798345617442510/58863869" ], [ "143543240122282594/9", "54310424664781703112741884/27" ], [ "21932292098978439904820/3568321", "3219209126882113291254038149097510/6740558369" ], [ "130893338118852930/361", "-30597161802530398523394340/6859" ], [ "414053988634420", "-2626948398704820269210" ], [ "478311460656900", "-67395889520080308010" ], [ "20822298416231551497220/43943641", "100561680095391510410886047503310/291302396189" ], [ "995281150564688360980/1907161", "-3697510664774461251644234425970/2633789341" ], [ "474209115686009700/841", "-75118812688062079506219410/24389" ], [ "775816978888420", "-12149729763811122725210" ], [ "23783724072221805/16", "-3176744871249302298563455/64" ], [ "7460889361050978666716020/26224641", "-20379053492046522649000488415619494810/134296386561" ], [ "1753748568441128308/1089", "2054491546835926432213605734/35937" ], [ "4980692083756696180/7569", "-4595470418622623132327656630/658503" ], [ "467596474010420", "-611791578487052205210" ], [ "3097490471898420", "-166616961664972326125210" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=389/10: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:02:11", "updated_at": "2026-07-01 22:41:37" }, { "id": 191, "curve_key": "2924316618320599734297591195298561:-157624488462778219456406517610332023994452455309441", "ainvs": [ "1", "0", "0", "-60923262881679161131199816568720", "182435750535622934108717118205361600171372294400" ], "rank_lower_bound": 16, "naive_height": 231.17510666228486, "faltings_height": 17.082959594776295, "conductor": "28964385051890469693884372342725226899763554916835745467259598596770", "bad_primes": [ "2", "3", "5", "7", "11", "13", "19", "23", "29", "37", "73", "5413", "33361703", "2721505543331", "57333637865117140073296578053387" ], "discriminant": "93858557525002069986716442228582690235767407598682422890314976013836608666063661484178636800000", "regulator": "4654554321845412.897032645881460013318440912154139638548237775729768145", "points": [ [ "948120211437022752", "-923167350663906647619020688" ], [ "21435739948266173578080/67081", "-3137468248676342929871311536556080/17373979" ], [ "243859843230968400", "-120362332081758768476537280" ], [ "104222286079490939040/5041", "-997050253259905842053863275120/357911" ], [ "2739289250202526800/169", "-4082763666617324166773111760/2197" ], [ "143889931307632230060/37249", "494799577777326710719500807360/7189057" ], [ "4294930541891040", "834638007028753640880" ], [ "107349527602354836/25", "-306438416685685033765056/125" ], [ "6556285940498177654880/1560001", "-48678867191471538240064329846480/1948441249" ], [ "104506446187837900/9", "27543714424019930053480540/27" ], [ "12320874007946940", "1141124473343389305468480" ], [ "13284156751610400", "1310481222233810387940720" ], [ "103260000815504640", "33089491507442908457627280" ], [ "1109432993238331471530", "-36953162831912838804179582295510" ], [ "5105061389726048340/1849", "14932639441279648341391124160/79507" ], [ "12386226484516320", "1152433950992307113428080" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-1146, -2304, -654, 3054, 2880, -1830] with shift t=1295/6: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:02:58", "updated_at": "2026-07-01 22:41:38" }, { "id": 235, "curve_key": "291537614333703744:-151865295230477283557887872", "ainvs": [ "0", "0", "0", "-6073700298618828", "175770017627867226340148" ], "rank_lower_bound": 15, "naive_height": 120.64183628862605, "faltings_height": 7.97610173327405, "conductor": "690285168700675717325304943706298441927480", "bad_primes": [ "2", "3", "5", "37", "61", "1061", "11174227", "71657492841305578040354317" ], "discriminant": "993016632286044059915490679818132686619195628800", "regulator": "1930058946037.1760717909398678090108062436141414502209083178458576916", "points": [ [ "33929674", "93551911110" ], [ "55028554", "90428617530" ], [ "51848404", "15497923230" ], [ "51767374", "8936997390" ], [ "20389894", "245774216430" ], [ "34564594", "84437781030" ], [ "53046094", "53382700530" ], [ "26873404", "-178763660730" ], [ "-29739926", "574541160390" ], [ "65790004", "246863819070" ], [ "-36266726", "590205480390" ], [ "-6254546", "462075298830" ], [ "70278844", "309891494970" ], [ "31681258", "123069479994" ], [ "57653926", "131291923986" ] ], "submitter": "RoyManami", "commentary": "Rank-15 naive-height record: h=120.642 (beats 127.132). New 3-parameter sub-family of Mestre locus 12*p5=5*p2*p3: six roots = union of two depth-2 cubics with (p-r)(q-s)=0 (equal product), beyond Mestre (u,v) slice. Sextuple {0,7,133,136,430,434}, T=162; base-point injection + Neron-Tate independence (15 indep).", "created_at": "2026-06-26 01:21:19", "updated_at": "2026-07-01 22:41:38" }, { "id": 86, "curve_key": "2536381595032617889:166595742322888303250363663", "ainvs": [ "1", "0", "0", "-52841283229846206", "-192819150610486916434364" ], "rank_lower_bound": 15, "naive_height": 127.13181051112655, "faltings_height": 8.662454551552132, "conductor": "75863584261726271948809574125392066219010", "bad_primes": [ "2", "5", "7", "19", "896405796700345079", "63632217828750709343" ], "discriminant": "9426722265250752028761189313525609860226811622809600", "regulator": "791888638835.30221384169538421040548888858303566962292163142474760247", "points": [ [ "-86289844", "1929897590322" ], [ "232261116", "252042240242" ], [ "-19704724", "-916915922718" ], [ "-219385604", "-916819471118" ], [ "-226833604", "-349143487278" ], [ "-223779924", "652546002082" ], [ "-63014844", "1699073084522" ], [ "-49541412", "1517727177010" ], [ "649349116", "15468856647442" ], [ "-224413004", "-603000225518" ], [ "-4649124", "-229662735118" ], [ "-15204004", "-779136381518" ], [ "-85738692", "-1925430491150" ], [ "-121604004", "-2105804614478" ], [ "-133371844", "2117209895922" ] ], "submitter": "David Renshaw", "commentary": "New curve of rank >= 15 and naive height 127.13, found as the specialization T = 1043/2 (t = 1043/4) of the 2-parameter Mestre rank-12 family over Q(t) used by Fermigier in 'Une courbe elliptique definie sur Q de rang >= 22' (Acta Arith. 82 (1997), sextuple {0,55,314,378,1007,1036}), located by a staged Nagao-sum sieve over ~1.9M specializations; 15 independent points certified by positive-definite Neron-Tate height pairing.", "created_at": "2026-06-11 15:29:48", "updated_at": "2026-07-01 22:41:38" }, { "id": 68, "curve_key": "0:-40586013776330216163919368288", "ainvs": [ "0", "0", "0", "0", "46974552981863676115647417" ], "rank_lower_bound": 15, "naive_height": 131.74644205965896, "faltings_height": 9.036526080763208, "conductor": "238313731807359099208895797698912716157803778347364012", "bad_primes": [ "2", "3", "1201", "6539317", "32344001", "61641367" ], "discriminant": "-953254927229436396835583190795650864631215113389456048", "regulator": "6004179571973955.9594170979361628807478033739008721205590284254100767", "points": [ [ "134903368", "7030622192107" ], [ "192112402", "7352882361535" ], [ "225027274", "7639981722671" ], [ "225083527", "7640541103540" ], [ "258845527", "8019816949840" ], [ "323938954", "8998197347759" ], [ "396959044", "10465465248949" ], [ "493287703", "12923142427688" ], [ "618875539", "16852542091394" ], [ "800577328", "23666090147563" ], [ "832883668", "24994838653493" ], [ "900602302", "27882595405255" ], [ "1938425152", "85618874611435" ], [ "-3801659403949148854700590661/10563498456094496025", "20677120321600099128973780340089600900562/34332992588487460476311920125" ], [ "-45422308525362180658274681/138657052360580625", "177509449917854185325666317221372981128/51631286351368162487765625" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Noam D. Elkies, \"j = 0, rank 15; also 3-rank 6 and 7 in real and imaginary quadratic fields\", NMBRTHRY posting, December 30, 2009. This is the Mordell curve y^2 = x^3 + 46974552981863676115647417. The submitted witnesses consist of integral points on this curve plus two points obtained from integral points on the 3-isogenous curve y^2 = x^3 - 27K via the dual 3-isogeny.", "created_at": "2026-05-29 11:30:20", "updated_at": "2026-07-01 22:41:39" }, { "id": 218, "curve_key": "37168047451033625161:-159335239524567280443702326341", "ainvs": [ "1", "0", "1", "-774334321896533858", "184415786422239825540685056" ], "rank_lower_bound": 15, "naive_height": 135.18594337975105, "faltings_height": 9.293392679912326, "conductor": "13737286227434328030396852859772916879098470", "bad_primes": [ "2", "5", "7", "19", "127", "151", "128983", "596538014056881445633451959507" ], "discriminant": "15022333163587070509998639856259499737597891168610762500", "regulator": "1779978689411.8084114367753707899730103398541064027289010096742171086", "points": [ [ "979729130622811281/166926400", "-959240687447310035515063389/2156689088000" ], [ "10219787479/4", "-976013371455203/8" ], [ "4528339940910/3481", "7625308316765958588/205379" ], [ "391446893015/11449", "15397599381839218531/1225043" ], [ "54547127865/64", "-6135098080565721/512" ], [ "1269739826015/14884", "-19806885383250721019/1815848" ], [ "23849164131/25", "2214544291703631/125" ], [ "-438537424", "20968123525679" ], [ "7914992168785/1089", "-22109430657093676121/35937" ], [ "4642634492777710/208849", "316088285805959912636756/95443993" ], [ "2450475884560/3249", "1004729461991195431/185193" ], [ "110589032755840/22201", "1145551647025539063808/3307949" ], [ "179225324275/169", "51800960300555549/2197" ], [ "3796926254535/289", "7382233305884976321/4913" ], [ "12459533075735/3844", "42451145483808864851/238328" ] ], "submitter": "Edgar Costa", "commentary": "Specialization at T = -490 of the Mestre/Fermigier family y^2 = r(x,T) on the sextuple (a) = (1608, -870, -1080, -642, 1542, -558) [r = degree-<=4 remainder in p6(x-T)p6(x+T) = g^2-r, g monic degree 6, p6(x)=prod(x-a_i)]. Located by a Mestre-Nagao sieve of this family. Fifteen independent rational points were obtained from the low-height Mestre quartic cover together with 2-descent covers; rank >= 15 follows from the positive-definiteness of their 15x15 Neron-Tate height-pairing matrix, verified independently in Sage and Magma. Conductor by Tate's algorithm. Proven lower bound on the rank (no exact-rank/Selmer/BSD claim).", "created_at": "2026-06-25 21:53:09", "updated_at": "2026-07-01 22:41:39" }, { "id": 237, "curve_key": "79278735968380698064:-786858559855956143215675588288", "ainvs": [ "0", "1", "0", "-1651640332674597876", "910715925208661721534017824" ], "rank_lower_bound": 15, "naive_height": 137.67569204492898, "faltings_height": 9.446902766047197, "conductor": "5919531324154970229554784141330370952904285996920", "bad_primes": [ "2", "5", "13", "19", "23", "26049688981495204319463052901471444080726483" ], "discriminant": "-69948023501250722618129515710347501757574437396885241600", "regulator": "111257299159024.59816588442634390170741817967774122122164844181174948", "points": [ [ "772668450", "9789861575182" ], [ "682982520231464/143641", "-17263025154188599268920/54439939" ], [ "2306049782", "96774035268922" ], [ "150596914681/64", "51336000912478035/512" ], [ "681402367466/49", "560166370225180550/343" ], [ "2852440687", "139313169639822" ], [ "18862882604/25", "1212361838592208/125" ], [ "5397415438786/23409", "83394215197427708990/3581577" ], [ "78607545570", "22036301809608398" ], [ "28772514445/36", "2172923165525957/216" ], [ "-87485153086/841", "802024535920980970/24389" ], [ "17438427206/25", "1237612245267346/125" ], [ "1043724563974/441", "937999906284868750/9261" ], [ "16728707607069638/165649", "2163508515105097434676454/67419143" ], [ "390658744", "18030759283680" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥15. Mestre construction from the rational 6-tuple family a(u,v)={u,v,-u-v,-u(u+2v)²/((u-v)(2u+v)),v(2u+v)²/((u-v)(u+2v)),(u-v)²(u+v)/((u+2v)(2u+v))} at (u,v)=(-10,-1), cleared tuple [-840, -84, 924, 640, -343, -297], shift t=784/1. This family satisfies the Mestre S=0 degree-drop condition identically (two depressed cubics with equal sum of squares). Certified rank 15 via 15 independent points from a quartic point search + Néron–Tate height-pairing matrix.", "created_at": "2026-06-26 06:57:30", "updated_at": "2026-07-01 22:41:39" }, { "id": 194, "curve_key": "553199431556088331024:-13049502537014064849315850384832", "ainvs": [ "0", "-1", "0", "-11524988157418506896", "15103590903163497035477292096" ], "rank_lower_bound": 15, "naive_height": 143.2926056061889, "faltings_height": 9.755185635612015, "conductor": "10190033470666423893407916147715865053798375920", "bad_primes": [ "2", "3", "5", "13", "17", "29", "31", "37", "1213", "228054274471183", "20879094585820301449" ], "discriminant": "-575294894722450744401258092356189386715188315557054668800", "regulator": "14815863059320.462760657204460902116663737179414860667111264912466988", "points": [ [ "678134618389", "-558429614392973290" ], [ "773401598709/4", "-680050397559446355/8" ], [ "12239211117264/529", "-42381308250908588520/12167" ], [ "12544641506", "-1358179764705030" ], [ "3569890425248/361", "-6390921341055807816/6859" ], [ "5591956987984/729", "-12102145909255225720/19683" ], [ "2898632381921/1681", "1303373127022665840/68921" ], [ "548698001744/289", "39940953579607080/4913" ], [ "79509693083361/38416", "81170157156022503465/7529536" ], [ "62974323862784/29929", "67360935645521657880/5177717" ], [ "124565294906581/50625", "461664469936998565354/11390625" ], [ "194522633305405/58564", "1644752730648824672287/14172488" ], [ "5865379264", "386380504805240" ], [ "3564819679739280/1907161", "25251714986265769716936/2633789341" ], [ "157879158605/121", "63656031680098026/1331" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥15. Mestre–Fermigier construction from the integer 6-tuple a=[240, -1692, -996, 1260, 1776, -588] with shift t=2352/1: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 15 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:03:13", "updated_at": "2026-07-01 22:42:37" }, { "id": 239, "curve_key": "1073250816532434693568:-35158133924832719085472876696064", "ainvs": [ "0", "-1", "0", "-22359392011092389449", "40692284642676185167439218585" ], "rank_lower_bound": 15, "naive_height": 145.27493742540233, "faltings_height": 9.786107191203211, "conductor": "493331968419716917139720664887174469134762256768", "bad_primes": [ "2", "3", "7", "13", "29", "4868091091", "13751713379", "7271987713848853806487" ], "discriminant": "85738128226224787181909938997870961494015365496541683712", "regulator": "72348684025139.813690956329432821964132838122276072356655182051278552", "points": [ [ "10851060085/4", "-258818176155/8" ], [ "7558450841269/289", "20461735427020481412/4913" ], [ "36592676846697/1024", "-219509068333802137163/32768" ], [ "12690550357/4", "-328743630743307/8" ], [ "-21288072143/16", "-16699737845264319/64" ], [ "1634480901", "-92265088908044" ], [ "-2111327789", "-280158194286516" ], [ "19227744004225597/3261636", "-1986693395508881051828039/5890514616" ], [ "40409079012536/19321", "-148973071535977430301/2685619" ], [ "208147940321649/76729", "390225018276560012/21253933" ], [ "-134774415986415/139129", "-12863588512365650166380/51895117" ], [ "1921166487515869/822649", "-25976048739534448334052/746142643" ], [ "269290852319401/103041", "-342687181344661622924/33076161" ], [ "60150234401830765/21353641", "-764007110263969721150820/98675175061" ], [ "4103100327", "-134264379870296" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥15. Mestre construction from the rational 6-tuple family a(u,v)={u,v,-u-v,-u(u+2v)²/((u-v)(2u+v)),v(2u+v)²/((u-v)(u+2v)),(u-v)²(u+v)/((u+2v)(2u+v))} at (u,v)=(1,-3), cleared tuple [20, -60, 40, 125, 3, -128], shift t=213. Two depressed cubics with equal product (satisfies Mestre 2e5=e2e3 identically). Certified rank 15 via 15 independent points (quartic point search + Néron–Tate height-pairing matrix). Smallest-conductor rank-15 of the family.", "created_at": "2026-06-26 15:03:44", "updated_at": "2026-07-01 22:42:37" }, { "id": 148, "curve_key": "3870469430745412686529:-240806713918040522796980722307617", "ainvs": [ "1", "0", "0", "-80634779807196097636", "278711474435456966031090976016" ], "rank_lower_bound": 15, "naive_height": 149.12309476917358, "faltings_height": 10.103220220897253, "conductor": "38802091035979009024912187041902860425470", "bad_primes": [ "2", "3", "5", "7", "17", "47", "67", "5659", "33260111", "104050049", "176241660609479" ], "discriminant": "-3574107331245376801878879599917457045580677964135913881600", "regulator": "154592244929.63709013408067677511651316763074973030093853274937906167", "points": [ [ "5122119536", "8656400619572" ], [ "5190432200", "-3927406903204" ], [ "5051647736", "16929747656492" ], [ "5191391816", "-3952015295908" ], [ "5157438344", "5111117078108" ], [ "5211393686", "5113496460722" ], [ "5072480576", "-14436717901348" ], [ "-10311204664", "-117707218206628" ], [ "5692157144", "64458341314268" ], [ "-10319795512", "108662784210140" ], [ "5223474566", "6211821138242" ], [ "5232768830", "-7165433634934" ], [ "3778637144", "167251374704588" ], [ "6769000520", "-207478525394596" ], [ "6159961880", "125475448853516" ] ], "submitter": "Alexey Pozdnyakov", "commentary": "Generated with assistance from OpenAI Codex, based on GPT-5. Method: generalized Mestre/Fermigier quartic-family search, sieving Mestre (u,v) families by Mestre-Nagao score, rescoring survivors, exact-height triaging the best candidates, and certifying top score/height candidates with GP famcert. This hit came from family mestre_u5_vm2, specialization T=453/1. GP famcert certified rank >= 15. conductor record for rank 15: log N 93.459293 < 94.129755", "created_at": "2026-06-24 17:55:21", "updated_at": "2026-07-01 22:42:37" }, { "id": 192, "curve_key": "40874711982984207510769:-8196790483963389560660650305834697", "ainvs": [ "1", "0", "0", "-851556499645504323141", "9487026023034811912831219901121" ], "rank_lower_bound": 15, "naive_height": 156.19439560733912, "faltings_height": 10.872367536621327, "conductor": "562577168211394579270024943629595507978443710", "bad_primes": [ "2", "3", "5", "11", "19", "103", "109", "6211", "64376373383029", "19987699093727394221" ], "discriminant": "638730941704979414939006251941574649893567559880993486621081600", "regulator": "6081709767502.5731747914878712722800396843622123174768992444677519175", "points": [ [ "84305679874/9", "-41233185475744451/27" ], [ "36241602249954/13225", "-4073566388142236145363/1520875" ], [ "-14513645610267290/2047761", "-11411968604968034879033251/2930345991" ], [ "5854928580399234/1075369", "-2496576161553282539156589/1115157653" ], [ "2941002666", "-2647270682374113" ], [ "600692104935810/326041", "-524072430071294943344139/186169411" ], [ "-7819076214", "-3958198824213513" ], [ "-23145631818534/1681", "-297253625339500924833/68921" ], [ "7156924540410/49", "18790187467681606353/343" ], [ "2536220959236", "-4038798111452968923" ], [ "124435835730", "-42782323227585561" ], [ "41205269280306/961", "-214420931827589747943/29791" ], [ "21031135188", "-938135835012105" ], [ "42768585876114/2809", "-33673699761100504821/148877" ], [ "5156498703514/441", "-9838663942889014997/9261" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥15. Mestre–Fermigier construction from the integer 6-tuple a=[-125, -99, -26, -18, 125, 143] with shift t=593/6: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 15 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:03:03", "updated_at": "2026-07-01 22:42:38" }, { "id": 220, "curve_key": "664432953815666778691825:-549132701009999108371297164497895625", "ainvs": [ "1", "0", "0", "-13842353204493057889413", "635570255797456549699738037418817" ], "rank_lower_bound": 15, "naive_height": 164.58729639029724, "faltings_height": 11.594497704331756, "conductor": "12223008395520638435655400939715776894443327750088254536450", "bad_primes": [ "2", "5", "19", "2887", "20611660717", "216219438825740695335021821292755331032329" ], "discriminant": "-4756217026864990828082229613662203105165787694114341605223424000000", "regulator": "2299012980980795.1736773300246733262723740839069277093718024770082457", "points": [ [ "2514318013738/9", "3679561913702325293/27" ], [ "35031218931168258/170569", "5660014854257299777589923/70444997" ], [ "2920373190306/25", "3093840971560740139/125" ], [ "268737294840178/3249", "1392139062719172942527/185193" ], [ "77950760643761618/1129969", "3593459039902576022864833/1201157047" ], [ "1079868038566/25", "1359257684239148771/125" ], [ "5495981224266002/229441", "1959028068119665673087321/109902239" ], [ "-2334203386631278/44521", "327738821395223206667629/9393931" ], [ "-105872842798", "30238515461284999" ], [ "323582351918322", "-5820726913620344706361" ], [ "16001135916564622/961", "2024022591810443849120209/29791" ], [ "1116391929791206802/78961", "1179532702855065438160211519/22188041" ], [ "-70392120718", "35512938899164359" ], [ "277081493813293058/316969", "144594497776810957638668973/178453547" ], [ "2513014308877942346562/6869641", "-125977339136208237332006595248701/18005329061" ] ], "submitter": "Edgar Costa", "commentary": "Specialization at T = -281 of the Mestre/Fermigier family y^2 = r(x,T) on the sextuple (a) = (1608, -870, -1080, -642, 1542, -558) [r = degree-<=4 remainder in p6(x-T)p6(x+T) = g^2-r, g monic degree 6, p6(x)=prod(x-a_i)]. Located by a Mestre-Nagao sieve of this family. Fifteen independent rational points were obtained from the low-height Mestre quartic cover; rank >= 15 follows from the positive-definiteness of their 15x15 Neron-Tate height-pairing matrix, verified independently in Sage and Magma. Conductor by Tate's algorithm. Proven lower bound on the rank (no exact-rank/Selmer/BSD claim).", "created_at": "2026-06-25 22:06:54", "updated_at": "2026-07-01 22:42:38" }, { "id": 238, "curve_key": "1232388808397940921824433:-1368984847784535468478339539721326441", "ainvs": [ "1", "-1", "1", "-25674766841623769204676", "1584473203460742150597373002163815" ], "rank_lower_bound": 15, "naive_height": 166.41426565188104, "faltings_height": 11.625099081975636, "conductor": "95954989391736297575743447520151210094560387026536946", "bad_primes": [ "2", "7", "13", "23", "67", "614071", "2845615572511", "195793702136100389221771138343" ], "discriminant": "-1382730624930385367184147512181586170312381345872576399281800347648", "regulator": "91113336322005.277313711589725666999350825220758797042718744371374129", "points": [ [ "4686108375477/49", "-664183300877107717/343" ], [ "131018426453", "21671238900850445" ], [ "721984601739957/5329", "-9471872484708889421835/389017" ], [ "2694575359197861/2209", "-138723138874027700845101/103823" ], [ "190686676611340509/2247001", "-13804565079891464631408393/3368254499" ], [ "14987765522285/169", "-4920895085425106975/2197" ], [ "24227790657649/225", "-27903398879047177237/3375" ], [ "3819910387281869/42025", "-11330999487134645156863/8615125" ], [ "77611481861", "-7701501794569091" ], [ "2638222669560109/26569", "-16264401587478186117297/4330747" ], [ "271036990440985/289", "-4400877261188912276575/4913" ], [ "229833258657625885/2627641", "-12003197530901784628769351/4259406061" ], [ "16035175610211357/35344", "1918001943185220915702165/6644672" ], [ "1552119729577773/17161", "-3324258507528512596305/2248091" ], [ "121581860292421693/961", "-42393848878672419779626805/29791" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥15. Mestre construction from the rational 6-tuple family a(u,v)={u,v,-u-v,-u(u+2v)²/((u-v)(2u+v)),v(2u+v)²/((u-v)(u+2v)),(u-v)²(u+v)/((u+2v)(2u+v))} at (u,v)=(-7,-1), cleared tuple [-210, -30, 240, 189, -125, -64], shift t=1791/4. This family satisfies the Mestre S=0 degree-drop condition identically (two depressed cubics with equal sum of squares). Certified rank 15 via 15 independent points from a quartic point search + Néron–Tate height-pairing matrix.", "created_at": "2026-06-26 06:57:35", "updated_at": "2026-07-01 22:42:38" }, { "id": 193, "curve_key": "4191610887043617416996401:-7297431744382413519811983154967154601", "ainvs": [ "1", "0", "0", "-87325226813408696187425", "8446101555990886693844233482455625" ], "rank_lower_bound": 15, "naive_height": 170.08538205546094, "faltings_height": 12.167890735676696, "conductor": "2282558436678897130581398498347303110857569170", "bad_primes": [ "2", "3", "5", "7", "11", "17", "19", "23", "31", "41", "675304453", "3187920097", "48610151546889853" ], "discriminant": "11801171435734791180357853315012481673861744917479863238791703372800000", "regulator": "14786943101801.821816404552640176100785119072680650943651152261285874", "points": [ [ "748436483759858950/1292769", "572940742521140768169429475/1469878353" ], [ "2457354167530/9", "1901167349508952825/27" ], [ "1902067649424250/19321", "76122157752942148735825/2685619" ], [ "101254141450", "25340801083372075" ], [ "554406704154395050/5193841", "218571678161936581007716925/11836763639" ], [ "90352672879870/841", "425494653096663013775/24389" ], [ "109936667050", "13212042066356875" ], [ "606701606153755/5476", "4639386552942000078275/405224" ], [ "112298274550", "7470676593563575" ], [ "144223305244433530/1274641", "5382887165331282577854355/1439069689" ], [ "13557424351450/121", "-11006174098335477775/1331" ], [ "77564540933783050/734449", "-12614392494911029526074525/629422793" ], [ "1126143904425925/4624", "12671172546060460859525/314432" ], [ "347065469002", "141223434230660971" ], [ "1471166114339530", "-56427728311403536554005" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥15. Mestre–Fermigier construction from the integer 6-tuple a=[-138, 138, 162, -60, -90, -12] with shift t=1265/9: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 15 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:03:08", "updated_at": "2026-07-01 22:42:39" }, { "id": 4, "curve_key": "10070973336541588631748191975281:-23029049564574022388452675835133173395678625929", "ainvs": [ "1", "0", "0", "-209811944511283096494753999485", "26653992551590286206010035905960909459942897" ], "rank_lower_bound": 15, "naive_height": 214.16163044680687, "faltings_height": 15.87181609687165, "conductor": "164007785374008600847294301252891222936905925387461785510", "bad_primes": [ "2", "3", "5", "7", "11", "13", "19", "23", "31", "5451269", "1288878077393", "1443189199877", "39758801179649479" ], "discriminant": "284205064269551456641094292955182094002428922847801912953464227178522579040000000000000000", "regulator": "197339248536833.87234569574893649111280704605566198247249889501297495", "points": [ [ "501737473225534", "-6905875723539454906517" ], [ "506962472826784", "-7112110805804266149017" ], [ "723528240924034", "-15925174720190407416017" ], [ "812523671448034", "-19814153331319366764017" ], [ "-177569372855966", "7636180350046204211983" ], [ "-419463909011966", "6392016193970552855983" ], [ "-499080158224466", "-2656234672725490419017" ], [ "-489670250151966", "-3461360602620746284017" ], [ "-161285098583966", "-7503202055979337116017" ], [ "576026995032034", "9845125145592749267983" ], [ "569204876688994", "9573248090546012288143" ], [ "437566790688034", "4315766925018741755983" ], [ "383738756328034", "1627462454566927475983" ], [ "380556863867362", "1386435548400000375823" ], [ "373690874760034", "658176881176342211983" ] ], "submitter": "David Renshaw", "commentary": "Found by Mestre (1992). A historical rank ≥ 15 record, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:11:45", "updated_at": "2026-07-01 22:43:36" }, { "id": 36, "curve_key": "-1647274302837199025396910363003835619879:-2751301637983405898417110838514633830651832727715466959927581", "ainvs": [ "1", "0", "1", "34318214642441646362435632562579908747", "3184376895814127197244886284686214848599453811643486936756" ], "rank_lower_bound": 15, "naive_height": 278.33435940448464, "faltings_height": 21.2523609564019, "conductor": "8754566324589342390719388201154487417353298842735433399274068130", "bad_primes": [ "2", "3", "5", "7", "11", "17", "67", "89", "139", "211", "281", "431", "443", "577", "647", "977", "1613", "3863", "10567", "11923", "15361", "73277" ], "discriminant": "-4383177432172452900429407516665453934052428354033391150708350854907555007169944896851345195235905604109325415039062500", "regulator": "26441299746981574.028361312934843803735096979068358730331594461979348", "points": [ [ "-5955399047526089895", "-52619192486073556789679851928" ], [ "4996845231479851005", "58996807068911558580932032822" ], [ "-13155206566829859045", "-21360729170232157127198638028" ], [ "5982316535030750730", "60031451151089353115173694947" ], [ "3520990345094746477605", "208928587539794577855401843236822" ], [ "-149780582516304339030/289", "-276460561394858085500066301116014/4913" ], [ "5537764707520796477505/256", "485495189997228202630782626087287/4096" ], [ "10379720384859947873670/529", "1299762167535132813050160826170149/12167" ], [ "3607902715536254330407755/36481", "6876286874413935169019164791798028712/6967871" ], [ "575451914344737045120/18769", "145126149873708232941281933935318916/2571353" ], [ "-635343181823720560310/81", "-35955391603910411255538526302302/729" ], [ "385275433846822770303250/9", "239142619084196570639847351097336094/27" ], [ "11630065797764473356485380/349281", "41921913401808378926412528792277717237/206425071" ], [ "2130862087205394565011555/206116", "6377017660319811010371576798520557433/93576664" ], [ "361886218060793196368152192851/82159049956", "1377397812507014979022748799194701395570639671/23549577125088104" ] ], "submitter": "David Renshaw", "commentary": "Found by Dujella (2002). A curve of rank exactly 15, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:14", "updated_at": "2026-07-01 22:43:36" }, { "id": 244, "curve_key": "2149850488656:-2127344071957778880", "ainvs": [ "0", "0", "0", "-44788551847", "2462203786988170" ], "rank_lower_bound": 14, "naive_height": 85.18925824647027, "faltings_height": 5.131289175169005, "conductor": "97849916723105200970396784134696", "bad_primes": [ "2", "19937", "100585774481", "6099217213561621" ], "discriminant": "3131197335139366431052697092310272", "regulator": "14021721492.1716638850528837824332581766485430502825171168406532475", "points": [ [ "-15631379015/99856", "-2369204678362285/31554496" ], [ "4052355557/3481", "-253871365077038/205379" ], [ "5262768921/196", "381775442683333/2744" ], [ "-222001919/1764", "5789112700393/74088" ], [ "1393321", "1626336962" ], [ "-9695799134155739/91418546025", "2144515602814138618661134/27640854483388875" ], [ "5612388061/15625", "353280062993066/1953125" ], [ "180414", "-15939116" ], [ "8914742837/39601", "-485001428074178/7880599" ], [ "73674598589/167281", "-17863087725156374/68417929" ], [ "34408221678887385/1335975352336", "-56224932570097492293210595/1544179095145451584" ], [ "790421", "678887362" ], [ "337343392162/142129", "195171010842841522/53582633" ], [ "98575401/4", "-978671293277/8" ] ], "submitter": "Andrew Sutherland", "commentary": "Found by Noam D. Elkies in June 2026, using an improved version of the methods described in his JMM 2023 talk https://abel.math.harvard.edu/~elkies/Elkies_JMM23.pdf\r\n", "created_at": "2026-07-06 21:36:14", "updated_at": "2026-07-06 21:36:14" }, { "id": 103, "curve_key": "13317251764761:-41197554564671958525", "ainvs": [ "1", "-1", "0", "-277442745099", "47682424180908449" ], "rank_lower_bound": 14, "naive_height": 90.66024430735854, "faltings_height": 5.550007778495223, "conductor": "2373990384245974896168946136843946", "bad_primes": [ "2", "3", "5393", "367261", "66588804478631114025089" ], "discriminant": "384586442247847933179369274168719252", "regulator": "28502581647696.2529128952344337988779234394456525903918680609725709", "points": [ [ "-178921/4", "1959819867/8" ], [ "-2174", "219740875" ], [ "188765", "45039138" ], [ "197344", "24725263" ], [ "442471", "107245690" ], [ "574252", "278511265" ], [ "906415", "735008122" ], [ "1559116", "1844506579" ], [ "2988133/16", "3117289423/64" ], [ "5030386/25", "216919319/125" ], [ "22990075/36", "78094116187/216" ], [ "12142321/25", "20700070241/125" ], [ "5651834483/9604", "279265261887261/941192" ], [ "47714038450/80089", "6936645295043041/22665187" ] ], "submitter": "David Renshaw", "commentary": "Found by Zev Klagsbrun.", "created_at": "2026-06-24 00:00:13", "updated_at": "2026-07-01 22:43:37" }, { "id": 3, "curve_key": "107915141124336:-1148663886952028319000", "ainvs": [ "0", "0", "1", "-2248232106757", "1329472091379662406" ], "rank_lower_bound": 14, "naive_height": 96.98577276472152, "faltings_height": 5.986406653793755, "conductor": "36275332432131715984679943280544970923", "bad_primes": [ "1217", "381991", "78031093338905335441668500509" ], "discriminant": "-36275332432131715984679943280544970923", "regulator": "9265899869259.67785615968081224333780816023564040627142679366174481", "points": [ [ "-1715883", "367667174" ], [ "-1695357", "517854321" ], [ "-1684243", "581714825" ], [ "-1664989", "676084814" ], [ "-1647951", "747688353" ], [ "-1621373", "844007025" ], [ "-1611950", "874669467" ], [ "-1579846", "968593731" ], [ "-1557343", "1026493950" ], [ "-1524175", "1102427297" ], [ "-1494242", "1163010071" ], [ "-1459188", "1226017574" ], [ "-1373814", "1351013139" ], [ "-1366017", "1360736678" ] ], "submitter": "David Renshaw", "commentary": "Found by Mestre (1986). A historical rank ≥ 14 record, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:11:44", "updated_at": "2026-07-01 22:43:37" }, { "id": 196, "curve_key": "28083791162533682241:-155024078761605995745679880289", "ainvs": [ "1", "-1", "1", "-585078982552785047", "179426017231461870343844271" ], "rank_lower_bound": 14, "naive_height": 134.42675592508496, "faltings_height": 9.130304325350192, "conductor": "15241162564541931655288262509967985228690", "bad_primes": [ "2", "3", "5", "13", "17", "5693", "134599091459570158754656517308997" ], "discriminant": "-1089582079144626137486468331565725618132214569984000000", "regulator": "418268198207.917346994581019827048869512072621751054302130902941757", "points": [ [ "745371953239691/2088025", "12089821218503314681926/3017196125" ], [ "7029737347501871/18653761", "284338310384923385718546/80565593759" ], [ "840199390735491/1985281", "7718166572582854963326/2797260929" ], [ "15489085411221/32761", "17275921179556260504/5929741" ], [ "23487639989/49", "1035223960079142/343" ], [ "82305778619/169", "6945576670507158/2197" ], [ "384190709219/289", "205579825992367902/4913" ], [ "84104283089/49", "22312839290733942/343" ], [ "229101138094731/96721", "3306517985548651348914/30080231" ], [ "3176482211", "-174276552517506" ], [ "-30900100109/81", "13581992367211726/729" ], [ "-2023912553071/6889", "10323306496153093428/571787" ], [ "151595411", "9706343114094" ], [ "1191988196819/5329", "3006911455148593998/389017" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥14. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=267/2: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 14 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:03:24", "updated_at": "2026-07-01 22:43:38" }, { "id": 221, "curve_key": "85026209906584393825:-797096715342679514131865860625", "ainvs": [ "1", "1", "1", "-1771379373053841538", "922565642019656180361670031" ], "rank_lower_bound": 14, "naive_height": 137.70154706301042, "faltings_height": 9.361977051261617, "conductor": "1067108442606429289452124323915075023850", "bad_primes": [ "2", "3", "5", "11", "13", "19", "31", "43", "13921", "13030973", "225715883", "47972031521" ], "discriminant": "-11961746928396976832774801995263862052981467695744000000", "regulator": "24502377124.0149976223379472061761656606064202568292589204243053485", "points": [ [ "-993905311", "41247580093347" ], [ "-618959415", "42212245856707" ], [ "-135133735", "34051074286467" ], [ "523368265", "11782775553267" ], [ "546935705", "10832299311747" ], [ "598100585", "8777953744707" ], [ "609494201", "8326581312963" ], [ "687993765", "5432840767267" ], [ "772324505", "3893973423747" ], [ "794554505", "4089081687747" ], [ "827810585", "4844491588707" ], [ "919786469", "8450622415863" ], [ "3291849305", "175391699574147" ], [ "9056167865", "852998874939267" ] ], "submitter": "Edgar Costa", "commentary": "Specialization at T = -309/2 of the Mestre/Fermigier family y^2 = r(x,T) on the sextuple (a) = (-44, -60, -6, 110, 94, -94) [r = degree-<=4 remainder in p6(x-T)p6(x+T) = g^2-r, g monic degree 6, p6(x)=prod(x-a_i)]. Located by a Mestre-Nagao sieve of this family. Fourteen independent rational points were obtained via 2-descent covers; rank >= 14 follows from the positive-definiteness of their 14x14 Neron-Tate height-pairing matrix, verified independently in Sage and Magma. Conductor by Tate's algorithm. Proven lower bound on the rank (no exact-rank/Selmer/BSD claim).", "created_at": "2026-06-25 22:06:57", "updated_at": "2026-07-01 22:43:39" }, { "id": 197, "curve_key": "760704094801311927921:-21402602511704044518245286166281", "ainvs": [ "1", "-1", "1", "-15848001975027331832", "24771530688804644612096284539" ], "rank_lower_bound": 14, "naive_height": 144.2821306343028, "faltings_height": 9.91865462217029, "conductor": "158660341771519260190542491906863061372490", "bad_primes": [ "2", "3", "5", "7", "17", "131", "6811583", "117316757", "141514035854442770479" ], "discriminant": "-10343870453933375939366089310754943399947700700823152640000", "regulator": "91751195039.7385955277149593413022755814531860544813979568131122369", "points": [ [ "-64560819529/25", "27521684792689179/125" ], [ "36123117509/25", "8740159303518117/125" ], [ "12682101068781/4225", "17895139115611159401/274625" ], [ "272865715950557/167281", "3906800984102749854129/68417929" ], [ "53671483231/9", "10186605986437651/27" ], [ "428560504493/49", "255573047630702943/343" ], [ "12449957567", "1325597777680281" ], [ "15851815757", "1938237977452041" ], [ "897117523463477/44521", "26382156112824679385091/9393931" ], [ "7586604431813/169", "20816879163190357437/2197" ], [ "-1901953423", "219166447716621" ], [ "-313569788467/289", "991035603369385113/4913" ], [ "-2964269157/4", "1520192145743473/8" ], [ "2016208521369317/885481", "18467414160521026345821/833237621" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥14. Mestre–Fermigier construction from the integer 6-tuple a=[-138, 138, 162, -60, -90, -12] with shift t=1227/5: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 14 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:03:29", "updated_at": "2026-07-01 22:43:40" }, { "id": 195, "curve_key": "2612721184108241595489:-155993673871847613487707706193937", "ainvs": [ "1", "-1", "1", "-54431691335588366573", "180548233661579697703929158997" ], "rank_lower_bound": 14, "naive_height": 148.2547364882902, "faltings_height": 10.344882362862343, "conductor": "4372904911177417822247776105333672098390", "bad_primes": [ "2", "3", "5", "7", "11", "13", "19", "97", "101", "260763461711492729725756564097" ], "discriminant": "-3760865945124333458688931654370706160938454707422168471961600", "regulator": "249748452478.896612615657083847531400376102137891106663850206439465", "points": [ [ "207486257967/49", "-55292996762624536/343" ], [ "5474933974027/1089", "-6623280826452051464/35937" ], [ "3485922442500407/683929", "-106607557059189060962616/565609283" ], [ "1698397246847/289", "-1239319603387616376/4913" ], [ "6268503683", "-292677744446952" ], [ "6886225535", "-363685789290072" ], [ "227040190559/25", "-82465709962026408/125" ], [ "13210232763", "-1329222140213832" ], [ "633401232898343/44521", "-14202660571046573935992/9393931" ], [ "137061348627", "-50670858651120040" ], [ "322490875343", "183089421925893528" ], [ "85458871823", "24892801911837528" ], [ "19817742035", "2623939877550036" ], [ "18737544623", "2395674470375448" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥14. Mestre–Fermigier construction from the integer 6-tuple a=[-110, 6, 60, 44, 94, -94] with shift t=491/4: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 14 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:03:18", "updated_at": "2026-07-01 22:44:38" }, { "id": 214, "curve_key": "27798226688830213653489:884868389321601533372341172335287", "ainvs": [ "1", "-1", "1", "-579129722683962784448", "-1024153228236700825546996771869" ], "rank_lower_bound": 14, "naive_height": 155.03777755009628, "faltings_height": 10.98580287758095, "conductor": "80368938829765696602245459305620130547747310", "bad_primes": [ "2", "3", "5", "7", "11", "13", "73", "241", "353", "171161", "279750562966837790136450937" ], "discriminant": "11977921723624303283051037363414017941731155172995286983285145600", "regulator": "48717653573893598.9939311056194054995916665713640284929986128650103", "points": [ [ "-12938798579711275/561001", "106534889369693588972853/420189749" ], [ "-9165284686125091/664225", "1127734230889137040381743/541343375" ], [ "-8432396697289099/3916441", "3575196644528288928032739/7750636739" ], [ "-1618070764674451/112225", "78217983332927259567087/37595375" ], [ "-21563290651425/961", "24444873195912878927/29791" ], [ "-10810436653197/841", "50551889024771630845/24389" ], [ "-2622368593781/225", "6869114780706266423/3375" ], [ "-90158812763/9", "52438974274566587/27" ], [ "20952892100429/625", "64874709412740057417/15625" ], [ "100146583014325/729", "986499899603264866051/19683" ], [ "165696032027705/2209", "2017400938636708177491/103823" ], [ "594013394643365/17161", "10154350491931317657147/2248091" ], [ "1300448789723773/25281", "41247706265133603067519/4019679" ], [ "6535886916365997/1681", "528382422141653838127321/68921" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥14. Mestre construction from the rational 6-tuple family a(u,v) = {u, v, -u-v, -u(u+2v)²/((u-v)(2u+v)), v(2u+v)²/((u-v)(u+2v)), (u-v)²(u+v)/((u+2v)(2u+v))} at (u,v)=(-12,-3), giving a=(-12,-3,15,16,-27/2,-5/2) with shift t=-1161/88 (cleared: tuple (-24,-6,30,32,-27,-5), t=-1161/44). This family satisfies the Mestre S=0 degree-drop condition identically — it is a union of two depressed cubics with equal sum of squares. Certified rank 14 via 14 independent points from an integer quartic-point search + Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 20:54:02", "updated_at": "2026-07-01 22:44:38" }, { "id": 198, "curve_key": "72945936872713395584521:-19759029617750051108484067702196069", "ainvs": [ "1", "1", "0", "-1519707018181529074677", "22869247242133051599466107911241" ], "rank_lower_bound": 14, "naive_height": 157.93783730300783, "faltings_height": 10.97539098710795, "conductor": "219931200577530279341714779925105066004870", "bad_primes": [ "2", "3", "5", "7", "13", "17", "31", "67", "163", "2671", "19213", "272760893775113648578459" ], "discriminant": "-1311295550412181298956751711791068502351802457869713490554687500", "regulator": "57498099076.9160905642634205944578651803129223187865761115602274920", "points": [ [ "29428257775715296/235225", "-4827910858317241171052311/114084125" ], [ "388726833316031142/13256881", "90552776990454414052053489/48268303721" ], [ "7911228532252973/258064", "295806306648889730642983/131096512" ], [ "121528141873272/3721", "644932695710007776079/226981" ], [ "12410076961752/361", "23007869174866244481/6859" ], [ "35203694052", "3605240641688769" ], [ "47187126232", "7498420893276259" ], [ "84857410752", "22471059109025379" ], [ "109029826357", "33959748881502634" ], [ "1070008775432", "1106105206155389859" ], [ "1212557108788/9", "-1284613342747699957/27" ], [ "37891831182", "-4437305361723291" ], [ "33961043607", "-3229175523309741" ], [ "28478113232", "-1639127433627741" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥14. Mestre–Fermigier construction from the integer 6-tuple a=[-756, 60, 210, 486, 654, -654] with shift t=1788/5: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 14 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:03:35", "updated_at": "2026-07-01 22:44:38" }, { "id": 35, "curve_key": "81230916433275311958757899750976:-725223800598609189906582606399776710024969039360", "ainvs": [ "0", "1", "0", "-1692310759026568999140789578145", "839379398840982294584587970773038145228669599" ], "rank_lower_bound": 14, "naive_height": 220.42454612632937, "faltings_height": 16.231295058121088, "conductor": "6116310391204554948397210171493960789901744808779000924864", "bad_primes": [ "2", "3", "7", "23", "43", "61", "79", "103", "157", "167", "179", "191", "199", "227", "229", "257", "271", "307", "487", "619", "1283", "3739" ], "discriminant": "5815704199930544426733052300699384236781490371077607532987271868542017804622894749291937792", "regulator": "1234880237432417.61331645290052732159258498172313923210000459986011", "points": [ [ "988927329611391", "11530632350958995292984" ], [ "681925764652050", "1568465432141985518043" ], [ "982479697930607", "11183605599461162465976" ], [ "665315262684159", "2820867968019861832248" ], [ "816133831449855", "1353961721252033023032" ], [ "20572279820626749/25", "250389733064763724244532/125" ], [ "681455320541454489/841", "8773166366975333247857352/24389" ], [ "8076800853180114", "716978870073714829212603" ], [ "7335280971697741/9", "33055815750694859233124/27" ], [ "457957201387226901/529", "58522059114298646792700636/12167" ], [ "127856380845197460021/625", "1445688913020859355673024573444/15625" ], [ "435462740765918268153/157609", "8219252385969460300109469047112/62570773" ], [ "811768719821415721233/18769", "23118208949065563006261517438272/2571353" ], [ "299475331388281185601/345744", "983394088087157723809198920329/203297472" ] ], "submitter": "David Renshaw", "commentary": "Found by Fermigier (1996). A curve of rank exactly 14, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:13", "updated_at": "2026-07-01 22:44:39" }, { "id": 158, "curve_key": "90075516496:-28257386435060120", "ainvs": [ "0", "-1", "1", "-1876573260", "32705933898332" ], "rank_lower_bound": 13, "naive_height": 75.76026257010892, "faltings_height": 4.245237096221498, "conductor": "13048475524065056885495472921", "bad_primes": [ "3", "4349491841355018961831824307" ], "discriminant": "-39145426572195170656486418763", "regulator": "740620279.4326354740651118477378116431858161004526275963581006162", "points": [ [ "7741928124496/345030625", "8823560993548208669/6408943859375" ], [ "132552615/3481", "-833270007898/205379" ], [ "1625667815/63001", "19131107316838/15813251" ], [ "15572079543/398161", "1098982569620230/251239591" ], [ "419003945245/34082244", "-674555808769168789/198972140472" ], [ "181207744936/4566769", "44420007350544652/9759185353" ], [ "43390", "5741911" ], [ "937218533/21316", "-18462206221191/3112136" ], [ "426410835/19321", "-3835378260566/2685619" ], [ "97096629/6889", "1720725812756/571787" ], [ "559519/25", "-173060047/125" ], [ "784478154/14641", "-16426944244376/1771561" ], [ "-965919/25", "861795179/125" ] ], "submitter": "Andrew Sutherland", "commentary": "Found by Noam D. Elkies in June 2026, using an improved version of the methods described in his JMM 2023 talk https://abel.math.harvard.edu/~elkies/Elkies_JMM23.pdf", "created_at": "2026-06-24 23:45:36", "updated_at": "2026-07-01 22:44:39" }, { "id": 89, "curve_key": "313431658176:-169340252787849600", "ainvs": [ "0", "0", "0", "-6529826212", "195995662948900" ], "rank_lower_bound": 13, "naive_height": 79.41254153101784, "faltings_height": 4.53993711701948, "conductor": "38252643743425234347938185256", "bad_primes": [ "2", "2741", "300427", "5806620890419143251" ], "discriminant": "1224084599789607499134021928192", "regulator": "3016078366.978183471091966719699927484854209920272441966848140771", "points": [ [ "-104", "14024078" ], [ "-1032", "14238446" ], [ "2022", "13520378" ], [ "-2566", "14585414" ], [ "3616", "13131302" ], [ "-4471", "15003371" ], [ "-4574", "15025558" ], [ "5193/4", "109550501/8" ], [ "5258", "12720346" ], [ "-6047/4", "114783601/8" ], [ "6337", "12444703" ], [ "-6694", "15472762" ], [ "-11031", "16330459" ] ], "submitter": "David Renshaw", "commentary": "Smallest known conductor (N = 38252643743425234347938185256) elliptic curve of rank >= 13, announced by Noam D. Elkies in 'New elliptic curves of large rank and low conductor' (preliminary report, JMM Special Session on arithmetic geometry informed by computation, Boston, January 4, 2023). Witness points found by ellratpoints and certified independent by Neron-Tate height pairing.", "created_at": "2026-06-11 16:55:32", "updated_at": "2026-07-01 22:44:39" }, { "id": 106, "curve_key": "3722260391616:-5577129652814340480", "ainvs": [ "0", "0", "0", "-77547091492", "6455011172238820" ], "rank_lower_bound": 13, "naive_height": 86.83605669550433, "faltings_height": 5.250447092824487, "conductor": "2299130784996838096010931135016", "bad_primes": [ "2", "7", "23", "809", "301831", "2630231441", "2779338763" ], "discriminant": "11845121804303709870648317207602432", "regulator": "9909246672.730008824881154639352039095879849904101177616705416006", "points": [ [ "-14090", "86861110" ], [ "474", "80114006" ], [ "80026", "27599402" ], [ "92561", "8378233" ], [ "231918", "30730714" ], [ "273042", "75081338" ], [ "273617", "75640537" ], [ "303402", "104192438" ], [ "343974", "143105494" ], [ "533632", "342099562" ], [ "1293552", "1438962938" ], [ "2895824", "4905678694" ], [ "3041382", "5282367062" ] ], "submitter": "David Renshaw", "commentary": "Found by Zev Klagsbrun.", "created_at": "2026-06-24 00:01:26", "updated_at": "2026-07-01 22:44:40" }, { "id": 61, "curve_key": "0:-650210564945071955736", "ainvs": [ "0", "0", "1", "0", "752558524241981430" ], "rank_lower_bound": 13, "naive_height": 95.8476558607853, "faltings_height": 6.044960564190403, "conductor": "81553583866934729091924419700360328569", "bad_primes": [ "3", "190669", "15787747861309" ], "discriminant": "-244660751600804187275773259101080985707", "regulator": "671098787666.6903251127775215461325395089393390179049329444143585", "points": [ [ "-3635871/4", "-314862269/8" ], [ "-3627871/4", "-644755819/8" ], [ "-3436375/4", "-2754036385/8" ], [ "-856990", "-350938406" ], [ "-845232", "-385629747" ], [ "-3332823/4", "-3338220641/8" ], [ "-792924", "-504008563" ], [ "-783172", "-521721587" ], [ "-2967655/4", "-4693365385/8" ], [ "-2538855/4", "-5639045385/8" ], [ "-521974", "-781244603" ], [ "-120210", "-866499531" ], [ "894450", "-1211674470" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Noam D. Elkies, 2009 rank-13 Mordell curve. Elkies originally gave the model y^2 = x^3 + 48163745551486811536 with 13 independent integral points; this database stores the global minimal model y^2 + y = x^3 + 752558524241981430, with the witness points transformed accordingly. Andrew Sutherland's MIT rank-record page lists the original curve and points.", "created_at": "2026-05-29 04:04:59", "updated_at": "2026-07-01 22:45:37" }, { "id": 200, "curve_key": "58267970912074998321:-479842416907924192177482963369", "ainvs": [ "1", "-1", "1", "-1213916060668229132", "555373168020983867224329231" ], "rank_lower_bound": 13, "naive_height": 136.68651052546213, "faltings_height": 9.347716627138887, "conductor": "6177584096452092230279206747860870", "bad_primes": [ "2", "3", "5", "7", "11", "31", "28755686340139143649765892789" ], "discriminant": "-18761498098102556371487217190377845102127324807168000000", "regulator": "12746195036.17266378810682497049084448405859392226472703342881533", "points": [ [ "14353961139/25", "862997361513753/125" ], [ "52759158221/25", "10747244403736929/125" ], [ "1986614540666021/638401", "83677707197040577478091/510082399" ], [ "4289402483179541/1324801", "266593073543863450615179/1524845951" ], [ "1225643375129621/323761", "41279856407830476915741/184220009" ], [ "33314611142315909/8334769", "5872502213681351200460547/24062478103" ], [ "4092206741", "253210335472629" ], [ "1271021187149/289", "1392062438251856277/4913" ], [ "59579399432021/11881", "449676106276783198641/1295029" ], [ "1607845076494709/299209", "63218916112170048001407/163667323" ], [ "5586961541", "410077035801429" ], [ "6838692021", "558641128118069" ], [ "341112840993869/5041", "6299253437563069859739/357911" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥13. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=2289/5: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 13 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:03:44", "updated_at": "2026-07-01 22:45:37" }, { "id": 201, "curve_key": "97584912720148833729:-960629570668111379574769901217", "ainvs": [ "1", "-1", "1", "-2033019015003100703", "1111839781374124406036388487" ], "rank_lower_bound": 13, "naive_height": 138.0817637177321, "faltings_height": 9.328045196898401, "conductor": "38936103625318215160524810514628790", "bad_primes": [ "2", "3", "5", "7", "13", "19", "37", "41", "109", "463", "1295809", "48684511", "51807199" ], "discriminant": "3746479851109552777476494863795034246987040897069772800", "regulator": "88401680947.32886924606362766775259075928101595166417470536057740", "points": [ [ "59588989675", "-14542085792666722" ], [ "38882274373", "-7661970975095272" ], [ "19682641375", "-2754329599786054" ], [ "9417396553", "-903979327498222" ], [ "3767050411", "-216589785741346" ], [ "30967318693/16", "-4258458933181603/64" ], [ "47002945240745371/35772361", "-5697237047256759529124786/213954491141" ], [ "1094757112585/961", "-493347650698187044/29791" ], [ "121707573635009/112225", "-507973989441742763468/37595375" ], [ "3615378856393/3481", "-2256266340234165578/205379" ], [ "23288579411/25", "-641863748313274/125" ], [ "108257172103/121", "-4013446770576302/1331" ], [ "749728245355/841", "-68704246405359218/24389" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥13. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=939/1: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 13 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:03:49", "updated_at": "2026-07-01 22:45:38" }, { "id": 203, "curve_key": "331074711495990018937:-6146034258883033837093628371789", "ainvs": [ "1", "0", "1", "-6897389822833125395", "7113465576836136307696346462" ], "rank_lower_bound": 13, "naive_height": 141.7867196553794, "faltings_height": 9.711030945984817, "conductor": "2840704603917455365782151764722058186", "bad_primes": [ "2", "3", "7", "13", "17", "19", "23", "37", "131", "177023219", "816206070761378053" ], "discriminant": "-859076487022818729321026088704523026210935982019962548656", "regulator": "2084640105.287431943944479547655984090376395520069190176449159295", "points": [ [ "1821696636986571/1681", "-77752259635965980361424/68921" ], [ "2651704780659/100", "-4297661292794605493/1000" ], [ "11101674264", "-1139650601752094" ], [ "6952530720", "-543352700813627" ], [ "10172211834471/2209", "-28051427564651955172/103823" ], [ "2670689975", "-87987691259172" ], [ "15395621611/9", "-483250003213552/27" ], [ "26665787397/16", "-1006638459099257/64" ], [ "1553029431", "-12140319849908" ], [ "248019741134157063/172318129", "-29292387555156537782663404/2262020079383" ], [ "208894998531/9409", "-76143106562997552752/912673" ], [ "7618219107", "629843965912912" ], [ "71190258388521651/5313025", "18654730438432478327441884/12246522625" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥13. Mestre–Fermigier construction from the integer 6-tuple a=[-138, 138, 162, -60, -90, -12] with shift t=296/9: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 13 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:04:00", "updated_at": "2026-07-01 22:45:38" }, { "id": 202, "curve_key": "369566474541430613329:-7136960510098350839243221718617", "ainvs": [ "1", "0", "0", "-7699301552946471111", "8260370960120371675267448841" ], "rank_lower_bound": 13, "naive_height": 142.0856795533432, "faltings_height": 9.672248385828121, "conductor": "377567090887779694323895274233922490", "bad_primes": [ "2", "3", "5", "7", "11", "13", "137", "91773698536932579421526229059" ], "discriminant": "-266808849431496166419387240071765982358706830625053900800", "regulator": "6519710445.716869994962627499550559448523343955253137275995743738", "points": [ [ "38654232246", "-7580643190350363" ], [ "203345746649964702/22819729", "87698553847271448902404101/109009845433" ], [ "6220241216550/49", "15509840583005224947/343" ], [ "110996977707102/82369", "430829851308717157515/23639903" ], [ "3833518434406794/2122849", "48470044901113166359581/3092990993" ], [ "9919516408158", "-31241782325935811763" ], [ "143201575382181750/1229881", "-54175148608867446972558159/1363938029" ], [ "1006894711087356/32041", "-31829950172644810031097/5735339" ], [ "11127316314114/625", "-36692149724825773563/15625" ], [ "1893419200260/169", "-2532103404406521687/2197" ], [ "51182275986198/6241", "-347474912239071226197/493039" ], [ "370092489225438/72361", "-6237092781954126380967/19465109" ], [ "4632706890", "-268365492663219" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥13. Mestre–Fermigier construction from the integer 6-tuple a=[-72, 120, 144, -48, -24, -120] with shift t=1516/7: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 13 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:03:54", "updated_at": "2026-07-01 22:45:38" }, { "id": 51, "curve_key": "-4338703069481731256479120080:0", "ainvs": [ "0", "0", "0", "90389647280869401176648335", "0" ], "rank_lower_bound": 13, "naive_height": 190.91211894712723, "faltings_height": 13.977583657235426, "conductor": "261449226737919396153756025756944483500777736264711200", "bad_primes": [ "2", "5", "7", "23", "43", "83", "97", "101", "269", "461", "503", "4013", "12829" ], "discriminant": "-47264606773393166969536680533142442422985196847451054883198340996443105471704000", "regulator": "138756719389405.4255918300409526821899187152598720710262995061562", "points": [ [ "109267165625", "3142916660550827000" ], [ "699012151740", "7970257947656823630" ], [ "1422638553735", "11466083720951564040" ], [ "3291776981199", "18254084011425711708" ], [ "5654458993409", "26303878853294220788" ], [ "7605531940860", "33576727388672968290" ], [ "7907148183375", "34772156979673232700" ], [ "9195899733104", "40110615271804927948" ], [ "13086347270535", "58514440970809001640" ], [ "22835291094000", "118201304172014979300" ], [ "27155267727809", "149930257416935566012" ], [ "35526638964665", "219205169249540459320" ], [ "48336648553881", "342497307164478802176" ] ], "submitter": "David Renshaw", "commentary": null, "created_at": "2026-05-28 20:49:27", "updated_at": "2026-07-01 22:45:39" }, { "id": 52, "curve_key": "135721397465823485897614042416:0", "ainvs": [ "0", "0", "0", "-2827529113871322622866959217", "0" ], "rank_lower_bound": 13, "naive_height": 201.2412052421559, "faltings_height": 14.838340848487816, "conductor": "28426385385919811326943632130923115672884025713568899872", "bad_primes": [ "2", "3", "11", "37", "41", "73", "151", "269", "509", "673", "1193", "2969", "5233" ], "discriminant": "1446775781054662070702764737544843998787085870189239433227587805462261153666449364032", "regulator": "108176633204870.8595106109963768445277899866191644600170784154096", "points": [ [ "-53023103285625", "29199177719416762200" ], [ "-52826063038023", "44172308695386224232" ], [ "-52264238470308", "70825438517010888582" ], [ "-29162438222148", "240117703609461349518" ], [ "-27920725708647", "239124633951324498624" ], [ "-23843884093425", "232084669370223457260" ], [ "-20893472520447", "223508666443210130124" ], [ "-19924665190425", "220062792377751438960" ], [ "-11023069423473", "172709750574634889268" ], [ "-9645751205607", "162407501940473002224" ], [ "-7293017369343", "142243865194633256268" ], [ "303030650052889", "5193240002220574777616" ], [ "2659109867774031", "137093723243731091154492" ] ], "submitter": "David Renshaw", "commentary": null, "created_at": "2026-05-28 20:49:28", "updated_at": "2026-07-01 22:45:39" }, { "id": 199, "curve_key": "999369977481478619918408638878049:-31737257213015815716596901650851489133526221191057", "ainvs": [ "1", "0", "0", "-20820207864197471248300179976626", "36732936589138673862895758597955508398047757956" ], "rank_lower_bound": 13, "naive_height": 227.96315152173966, "faltings_height": 16.82881768633617, "conductor": "21785392458764315483988614758901932764423833726404496768609056369259382575196330", "bad_primes": [ "2", "3", "5", "7", "13", "19", "86044359449701746413144681", "4881201479084572277272794646126863424004819546339" ], "discriminant": "-5290724783356970045570091261064369041862691109828850503901233859803837652758778996437693235200", "regulator": "3144017645707793.395706961347052974878979749568459122697271441584", "points": [ [ "1162634353995497656800804/7778521", "1253039186201942747498661935105222982/21694295069" ], [ "3627637297190869531/36", "6902364641611559233412383477/216" ], [ "1558179052657862437/144", "1794466997628157186816731475/1728" ], [ "3881968959990696", "120039898039176624614682" ], [ "2837185351207296", "22369929495303810169122" ], [ "133369750392561176/49", "5181060484570817333704274/343" ], [ "290049012608047521903204/131859289", "60156526711202358713019503533166394/1514140215587" ], [ "27074001988877942159/6724", "-74342839994965136661541523599/551368" ], [ "2480181137075511944/289", "-3439495665410956315384662434/4913" ], [ "9363687175740228", "-814109553204195618921090" ], [ "4354296816133898977943076/338376025", "-8579048965788332093253582643648116486/6224426979875" ], [ "11397032844750127922575524/4282000969", "3685010112080220968145792764888880246/280201297408453" ], [ "19369629507101316", "-2626899353324391553849218" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥13. Mestre–Fermigier construction from the integer 6-tuple a=[240, -1692, -996, 1260, 1776, -588] with shift t=2789/9: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 13 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:03:40", "updated_at": "2026-07-01 22:46:36" }, { "id": 45, "curve_key": "26914364817703967532575697223238202217921:-4578705818068316799984129290405535142567907177009541961541281", "ainvs": [ "1", "0", "0", "-560715933702165990261993692150795879540", "5299428030171662962897867758309003693598430128674403539600" ], "rank_lower_bound": 13, "naive_height": 279.3530439307271, "faltings_height": 21.202075126528637, "conductor": "37782801949200843416322963769043529248461782217859568090", "bad_primes": [ "2", "3", "5", "7", "11", "13", "17", "19", "23", "29", "31", "37", "41", "43", "47", "53", "59", "61", "113", "197", "317", "3313949", "2831657657", "4864617187" ], "discriminant": "-849674615711570771215349016224804046891153693433134502714153993581965535857133434327195120355094716597801586176000000", "regulator": "473362182709.7477529528683850131310786537924858078223167486334695", "points": [ [ "30978765264695144680", "132887030666136666891811891660" ], [ "42714959134973152720", "243484714136201166251098454860" ], [ "17517044808100174120", "29195754303760059255051062860" ], [ "14407536544275712180", "14545360491234198754332073660" ], [ "27024189809840210680", "99410687206608213450615379660" ], [ "15223373261754620200", "17072594325815595384787766860" ], [ "-19625425810319741720", "93513850432355415588023731660" ], [ "18799275311571587070", "37447008534737807545137468570" ], [ "-18348119874544858160", "97008001583008247645310994660" ], [ "16004432727240519730", "20612929621400491642269765910" ], [ "-27235206268166134070", "19202914916421997002398631160" ], [ "13326569619771689680", "13920236659741078875971506660" ], [ "27069452184006616180", "99781951498337473669562238160" ] ], "submitter": "David Renshaw", "commentary": "Rank 13 curve due to Eroshkin (2007), listed on Dujella's high-rank elliptic curves with prescribed torsion page for torsion group Z/3Z: https://web.math.pmf.unizg.hr/~duje/tors/z3old8910111213.html. The model is y^2 + x*y = x^3 - 560715933702165990261993692150795879540*x + 5299428030171662962897867758309003693598430128674403539600, with the 13 listed integral independent points.", "created_at": "2026-05-28 05:18:56", "updated_at": "2026-07-01 22:46:37" }, { "id": 157, "curve_key": "10910016201:-762285607502085", "ainvs": [ "1", "-1", "0", "-227292004", "882331831684" ], "rank_lower_bound": 12, "naive_height": 69.33884136527462, "faltings_height": 3.811265423121493, "conductor": "12212741842431104280769438", "bad_primes": [ "2", "17", "8046817087", "44638555294561" ], "discriminant": "415233222642657545546160892", "regulator": "42564662.256038750630862983698271383495644865238969730938118940", "points": [ [ "117790/9", "-10607936/27" ], [ "1309577033/73441", "31285012576920/19902511" ], [ "1881346627/18769", "80581143046275/2571353" ], [ "2591120/961", "15977490022/29791" ], [ "-91616894/5625", "-213077007104/421875" ], [ "37120", "6583838" ], [ "15910", "-1145142" ], [ "13305206/841", "27031784336/24389" ], [ "-57830/49", "367859024/343" ], [ "34256", "-5787210" ], [ "17920627/1369", "19285307426/50653" ], [ "305109061/24336", "192990890929/3796416" ] ], "submitter": "Andrew Sutherland", "commentary": "Found by Noam D. Elkies in June 2026, using an improved version of the methods described in his JMM 2023 talk https://abel.math.harvard.edu/~elkies/Elkies_JMM23.pdf", "created_at": "2026-06-24 23:38:30", "updated_at": "2026-07-01 22:46:37" }, { "id": 88, "curve_key": "44771207376:-9549233983224000", "ainvs": [ "0", "0", "0", "-932733487", "11052354147250" ], "rank_lower_bound": 12, "naive_height": 73.590474670011, "faltings_height": 3.987756536788123, "conductor": "26145292874820119408329144", "bad_primes": [ "2", "125207", "26102067850459758049" ], "discriminant": "-836649371994243821066532608", "regulator": "58563136.762357434222764319524246397648310277411773547153070685", "points": [ [ "361", "3273482" ], [ "877", "3199222" ], [ "1425", "3118670" ], [ "-1927", "3583654" ], [ "2026", "3028358" ], [ "-2363", "3639122" ], [ "3001", "2877542" ], [ "-3261", "3749576" ], [ "3533", "2793046" ], [ "-4019", "3838762" ], [ "-4394", "3881488" ], [ "4394", "2653066" ] ], "submitter": "David Renshaw", "commentary": "Smallest known conductor (N = 26145292874820119408329144) elliptic curve of rank >= 12, announced by Noam D. Elkies in 'New elliptic curves of large rank and low conductor' (preliminary report, JMM Special Session on arithmetic geometry informed by computation, Boston, January 4, 2023). Witness points found by ellratpoints and certified independent by Neron-Tate height pairing.", "created_at": "2026-06-11 16:54:49", "updated_at": "2026-07-01 22:46:37" }, { "id": 1, "curve_key": "304790815056:-166878443731135320", "ainvs": [ "0", "0", "1", "-6349808647", "193146346911036" ], "rank_lower_bound": 12, "naive_height": 79.32867457792244, "faltings_height": 4.46765893551938, "conductor": "269601712590130409544942497797", "bad_primes": [ "60259", "550469", "11241887", "722983930261" ], "discriminant": "269601712590130409544942497797", "regulator": "37541965361.758320665649129943760568697141050295736574648657389", "points": [ [ "49421", "200114" ], [ "49493", "333458" ], [ "49513", "362258" ], [ "49632", "502899" ], [ "49667", "538049" ], [ "49797", "654674" ], [ "49899", "735713" ], [ "50012", "818375" ], [ "50165", "921837" ], [ "50215", "954017" ], [ "51108", "1454591" ], [ "-3659", "14708205" ] ], "submitter": "David Renshaw", "commentary": "Found by Mestre (1982). A historical rank ≥ 12 record, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 21:57:08", "updated_at": "2026-07-01 22:46:38" }, { "id": 105, "curve_key": "2231819105841:-3306892292536952745", "ainvs": [ "1", "-1", "1", "-46496231372", "3827433258938575" ], "rank_lower_bound": 12, "naive_height": 85.3015143341792, "faltings_height": 4.9649896933349895, "conductor": "1264178492449062735461483178", "bad_primes": [ "2", "3", "655657", "107117194614550556453" ], "discriminant": "104856020877695059530117260716032", "regulator": "703896103.94797884020123281580587091171576593109950500826130294", "points": [ [ "1009", "61485425" ], [ "33409", "48059585" ], [ "44917", "42751157" ], [ "74813", "27668797" ], [ "105061", "10053317" ], [ "107167", "8626907" ], [ "110005", "6562229" ], [ "111421", "5423237" ], [ "133741", "1003877" ], [ "141443", "8905789" ], [ "146971", "12905837" ], [ "234613", "76254389" ] ], "submitter": "David Renshaw", "commentary": "Found by Zev Klagsbrun.", "created_at": "2026-06-24 00:01:15", "updated_at": "2026-07-01 22:46:38" }, { "id": 205, "curve_key": "865501542262379153750461947664:-810356568344230487580803832135872788497253312", "ainvs": [ "0", "1", "0", "-18031282130466232369801290576", "937912694842853350198553539190914682552740" ], "rank_lower_bound": 12, "naive_height": 206.8120965287715, "faltings_height": 15.080140699542088, "conductor": "5883556108705604864069301892047856053057376997561594085263469332125323440", "bad_primes": [ "2", "5", "13", "17", "19", "23", "157", "241", "617", "41665714723", "46051797406879788477331351041654345220422732481" ], "discriminant": "-4824473724726869454399342069094860042611338540998910914446745354020291555955475532800", "regulator": "23816794754349.050264022329195680584976251521649799131276940920", "points": [ [ "1019510314870079278/9", "-1029407033233963549733458760/27" ], [ "898379318706026386/81", "-851447619821221046090639324/729" ], [ "65554377206988649/144", "-16124573198339556102902467/1728" ], [ "89943481345135872/841", "-11766652141274314098778430/24389" ], [ "558225231481187845883/7360369", "-1625706497883799859381407016320/19968681097" ], [ "288028626599061/4", "-908295957363279970007/8" ], [ "36286970506582", "-575667222416485993560" ], [ "17527214855180285974742/58507201", "2119063633757101595805966005518120/447521580449" ], [ "855715812309179", "24740771596287139505996" ], [ "987441988220392", "30755977676121784722870" ], [ "41176586031416048557327/31348801", "8313519254200782912639963937076910/175521936799" ], [ "17435634703211710010126/200647225", "-468872888856202121848899501476824/2842167942125" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=10852/13: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 12 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:04:09", "updated_at": "2026-07-01 22:46:38" }, { "id": 204, "curve_key": "1959173408006340874297187670049:-2743405664101790044589728974331337298125159857", "ainvs": [ "1", "0", "0", "-40816112666798768214524743126", "3175238037154846187302797412986141032805156" ], "rank_lower_bound": 12, "naive_height": 209.25105855284116, "faltings_height": 15.179948694345056, "conductor": "8207915399785028827911225140199659880348230953515903817573863786069230", "bad_primes": [ "2", "5", "7", "17", "31", "157", "1417178528658108860031670181482848799986917717765832284580576751" ], "discriminant": "-3623222374799711246641358701859136275557470213922538013251076487946923600073999257600", "regulator": "51826916814533.727636603725739418132382839118508075688787171474", "points": [ [ "9770649366740959079884/619369", "-965716610723945297620503113535234/487443403" ], [ "60233506353420124/9", "-14776159007078725940508866/27" ], [ "1632143825512484940100/3374569", "-60920056763901273997043006329398/6199083253" ], [ "28290587106843356/361", "4641255380757530029904158/6859" ], [ "110391515971620", "121441937340674347746" ], [ "19680454388442956/169", "80229753769020194626786/2197" ], [ "17387195299617109/144", "147427941541656072822491/1728" ], [ "150037170137836", "654838533101906691642" ], [ "244244363437608", "2788656466736583932670" ], [ "5484612139645565/16", "347390158672825672194889/64" ], [ "22903392289503081044364/49042009", "3184412100432593479561128806746734/343441189027" ], [ "419339596095967/4", "1761051162359423345641/8" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=787/7: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 12 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:04:04", "updated_at": "2026-07-01 22:47:35" }, { "id": 208, "curve_key": "14625378343497043567612061435329:-55953489917540816135135028393347911408859680417", "ainvs": [ "1", "0", "0", "-304695382156188407658584613236", "64760983700857400691050362439563148137462416" ], "rank_lower_bound": 12, "naive_height": 215.28169998638668, "faltings_height": 15.6797000963728, "conductor": "1107988077126197589769596508555487707559725219449598542068453069615896859730", "bad_primes": [ "2", "3", "5", "23", "31", "157", "4513", "11249219", "1500692293891444032300619619", "4330572287574020219420543749507" ], "discriminant": "-1388810570434782811007607331048073936871115674258540673227450623892810201037842324019200", "regulator": "147652040099442.05947295954239201960603808166881615690156257635", "points": [ [ "44278949867297760957064/935089", "-9316796137756519793643768214532204/904231063" ], [ "158955475956867724/9", "-63343723260583635566586476/27" ], [ "7690985079964216766320/5368489", "-630406900594136415745523463507692/12438789013" ], [ "77834314090789836/361", "20678751759625129232805588/6859" ], [ "299913011917936", "596149116860699711812" ], [ "53714858859344400/169", "350978976489095631299388/2197" ], [ "47471468885246317/144", "649332284048410916301671/1728" ], [ "399524613364744", "2607635597175266655628" ], [ "656000246676580", "12131836808507316807904" ], [ "15043833852323541/16", "1580027181914339609144013/64" ], [ "101292582467515835873416/80263681", "29562434295869282672212131124865908/719082318079" ], [ "1139227591149379/4", "8328224915694189545111/8" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=967/9: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 12 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:04:24", "updated_at": "2026-07-01 22:47:35" }, { "id": 206, "curve_key": "17955918111461417803920655156129:-73176126388097287413385955954358513712822872817", "ainvs": [ "1", "0", "0", "-374081627322112870915013649086", "84694590726964421851283320326657184990555716" ], "rank_lower_bound": 12, "naive_height": 215.89641765065846, "faltings_height": 15.917906104017392, "conductor": "431184740183886106535857306115602438132236071442111719243984779367694673530", "bad_primes": [ "2", "3", "5", "7", "19", "31", "157", "4517", "21447358407501679739", "229195053659697773915482188781052624679234107" ], "discriminant": "251453575815775763875321183856781238190717796098320943436148000167676873962388406215526400", "regulator": "128581143130572.87247198242309993116966404006654526604274397426", "points": [ [ "1311962006756715124/9", "1502720013154061565338633534/27" ], [ "1622964775848764290/81", "2066633657734054159603682216/729" ], [ "2072780699018820625/576", "2943546468634289028128158229/13824" ], [ "2186299717178498316/841", "3149957659531808431157396778/24389" ], [ "21543146691997293/16", "2877874286932028601295563/64" ], [ "25660979318382673047964/62520649", "-270616637356825393764260426373754/494350771643" ], [ "29542255198291148257780/69839449", "-854884756696283163043353677345782/583648275293" ], [ "36612916576974039331564/79798489", "-2214006161797201672783134021818326/712839902237" ], [ "539247600876988", "-6307035117458519888078" ], [ "561618735181336", "-7193514911777232202958" ], [ "29051022192219566367268/57623281", "2153653151864037845479171859549542/437418326071" ], [ "3094088113068047834300476/664041361", "5397713329122196810957917667676278178/17111681831609" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=7907/3: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 12 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:04:14", "updated_at": "2026-07-01 22:47:36" }, { "id": 207, "curve_key": "160382049829531714473522201612289:-2038251840626530708856061601274460704106400167937", "ainvs": [ "1", "0", "0", "-3341292704781910718198379200256", "2359087778502928782734938491871399298406363136" ], "rank_lower_bound": 12, "naive_height": 222.4723539266345, "faltings_height": 16.360697581894815, "conductor": "877030824470138245488472257225334285674448305325975329534240593693229985430", "bad_primes": [ "2", "5", "7", "13", "17", "19", "157", "1153", "931690673197751", "1040689837985892942452948677587572182667636492893" ], "discriminant": "-16816563281216520792334587550778720192475526428103707017379511232474985349250763174857420800", "regulator": "296423040996438.87421831030116833563744437043438121358288765559", "points": [ [ "10124204966610246124/9", "-32213715633639840416999764496/27" ], [ "8018456777604735778/81", "-22701924064273317034280911460/729" ], [ "258291942189627096/169", "-63006127372568334146060224/2197" ], [ "87243969374310784555424/78552769", "-2963040146283390716652256622504576/696213191647" ], [ "157375705483764349/144", "-6177158242078167048846049/1728" ], [ "987942511482496", "-4727008292132400886208" ], [ "13951748666332005/16", "-666805218776552253396811/64" ], [ "397195526444269578582656/177129481", "184755682663955878491031799200844608/2357416262629" ], [ "10083618951832736", "996976824896800215156352" ], [ "11918542857126748", "1286695214692376067191380" ], [ "6843911934657346548837664/416445649", "17797961083287660114813363226306796416/8498406359143" ], [ "2769496506371736", "-119782031103604577330648" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=8863/13: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 12 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:04:19", "updated_at": "2026-07-01 22:47:36" }, { "id": 209, "curve_key": "1421665827432625232798174635095841:-52984092323815152647441895199926527602915169571889", "ainvs": [ "1", "0", "0", "-29618038071513025683295304897830", "61324180930341609392295317188644303710246778852" ], "rank_lower_bound": 12, "naive_height": 229.0114121122855, "faltings_height": 16.955808744559274, "conductor": "28341832953461142174879989692232359137024812210317425503392357305906630092930", "bad_primes": [ "2", "5", "13", "17", "23", "31", "157", "6329", "694987", "462422851227531199", "20539201174156553881", "210944648998482573373" ], "discriminant": "38230734935601059553922446496293184814658591376828071969280861189829807789616058154834984960000", "regulator": "1760119853264209.7771865074582412243590679211918621886021916407", "points": [ [ "6103105912463294758/9", "-15076912576194046638463529894/27" ], [ "2721078832888637224/81", "-4432992434214499160854395398/729" ], [ "39890447057124135690412/1203409", "-7865840853234332855580384931422342/1320139673" ], [ "4306516726393068", "-116801695188176626466934" ], [ "586432636053005156/169", "-40007927829283736740469774/2197" ], [ "3414095341478124", "-480513742457692657782" ], [ "491831239431911341/144", "4806745636338001659536669/1728" ], [ "104843902700829269/16", "24671933214190445149476127/64" ], [ "845855098895920330636/100489", "20357579809465838424373863781554/31855013" ], [ "9980933142991084", "871779419766022779836298" ], [ "13084656463879404", "1383467833157765669272458" ], [ "7989839137371514742387916/387814249", "21864064605954559349916148628421134226/7637226005557" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=5485/16: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 12 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:04:29", "updated_at": "2026-07-01 22:48:33" }, { "id": 213, "curve_key": "3577957125609504224117667787414369:-210873622699844738147045972586854887727236925732497", "ainvs": [ "1", "0", "0", "-74540773450198004669118078904466", "244066692939635107384386902977544619850388014596" ], "rank_lower_bound": 12, "naive_height": 231.78030021311164, "faltings_height": 17.19719209430338, "conductor": "10465426762586441668769337684707368604491256699170701827399830238043087368190", "bad_primes": [ "2", "3", "5", "7", "11", "17", "19", "157", "6651361", "21871169", "117189223", "2264709407", "136116631176129536978806192189908923" ], "discriminant": "773452069750637843410919255151077621482402538193851004785076611405015441616047124089938250956800", "regulator": "545624411053203.73045398884583072180213372155149257502745762152", "points": [ [ "11421650765628186754/9", "-38599643258316838007145121946/27" ], [ "5657778683304010828/81", "-13359296359452931217293552370/729" ], [ "1859392181426721484548004/45873529", "-2482023862685444193093304134783782266/310701411917" ], [ "6670385051890276", "-208910006765036920615298" ], [ "953233885740561876/169", "-121770304938003450603963594/2197" ], [ "787993857911111269/144", "-9344055086171434479140719/1728" ], [ "5476037191636420", "9406440863363801143966" ], [ "154182032082462285/16", "41506127422013923347712539/64" ], [ "17650888545826756", "2104176405239656414694302" ], [ "22770019726862068", "3217519285772321257814830" ], [ "371978685111016429303396/10569001", "220592085961447243696290765616630922/34359822251" ], [ "10776150844881796", "-831979087315908337590818" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=6773/18: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 12 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:04:49", "updated_at": "2026-07-01 22:48:33" }, { "id": 50, "curve_key": "785235216:-22700890644120", "ainvs": [ "0", "0", "1", "-16359067", "26274178986" ], "rank_lower_bound": 11, "naive_height": 61.50685055014001, "faltings_height": 3.04119413905516, "conductor": "18031737725935636520843", "bad_primes": [ "18031737725935636520843" ], "discriminant": "-18031737725935636520843", "regulator": "28087940.2288126473904406024985250980292392927967906350917450", "points": [ [ "-4683", "13535" ], [ "-4657", "38196" ], [ "-3091", "217503" ], [ "-826", "198048" ], [ "1165", "93792" ], [ "1921", "44015" ], [ "2315", "28452" ], [ "2837", "51935" ], [ "3656", "123825" ], [ "5753", "350096" ], [ "9565", "863072" ] ], "submitter": "David Renshaw", "commentary": "Smallest known conductor of an elliptic curve with rank ≥ 11 (Elkies–Watkins, 2004). From their paper \"Elliptic Curves of Large Rank and Small Conductor\" (arXiv:math/0403374).", "created_at": "2026-05-28 18:41:24", "updated_at": "2026-07-01 22:48:33" }, { "id": 138, "curve_key": "1638031881:-60060810310245", "ainvs": [ "1", "-1", "0", "-34125664", "69523358164" ], "rank_lower_bound": 11, "naive_height": 63.65028385667353, "faltings_height": 3.274936832305714, "conductor": "227946110025657660240686", "bad_primes": [ "2", "151", "2581599337", "292372419289" ], "discriminant": "455892220051315320481372", "regulator": "98660495.1036795550880639813729964725582620483236327750338506", "points": [ [ "2482", "9188" ], [ "2438", "27270" ], [ "2408", "34950" ], [ "2370", "43012" ], [ "2360", "44938" ], [ "2234", "65458" ], [ "2086", "85044" ], [ "4200", "14342" ], [ "4218", "22642" ], [ "2008", "94350" ], [ "1950", "100958" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:47", "updated_at": "2026-07-01 22:48:34" }, { "id": 137, "curve_key": "1969555161:-91646329814685", "ainvs": [ "1", "-1", "0", "-41032399", "106082399089" ], "rank_lower_bound": 11, "naive_height": 64.29791608749225, "faltings_height": 3.293194564452344, "conductor": "219576020293485812169274", "bad_primes": [ "2", "9913609", "11074474507391093" ], "discriminant": "-439152040586971624338548", "regulator": "9786539.43783265571672432007979536225451355144124014541976368", "points": [ [ "3790", "68807" ], [ "3342", "77515" ], [ "3996", "74899" ], [ "4238", "88935" ], [ "2940", "102707" ], [ "2878", "107295" ], [ "4382", "99823" ], [ "2764", "116003" ], [ "2756", "116625" ], [ "2672", "123225" ], [ "2660", "124177" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:45", "updated_at": "2026-07-01 22:48:34" }, { "id": 136, "curve_key": "1828752681:-100110203830485", "ainvs": [ "1", "-1", "0", "-38099014", "115877816224" ], "rank_lower_bound": 11, "naive_height": 64.4745854668461, "faltings_height": 3.3833176859920684, "conductor": "66484354768372183177742", "bad_primes": [ "2", "17", "277", "25301", "279012130635919" ], "discriminant": "-2260468062124654228043228", "regulator": "5402298.84447187773578025736585607570884012602919864656010005", "points": [ [ "3370", "158778" ], [ "3710", "158098" ], [ "2826", "174010" ], [ "2622", "183088" ], [ "2010", "216748" ], [ "1398", "254930" ], [ "5478", "264722" ], [ "208", "328472" ], [ "-778", "381240" ], [ "-1424", "409664" ], [ "3795", "159152" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:44", "updated_at": "2026-07-01 22:48:34" }, { "id": 139, "curve_key": "2730287721:-145695022596405", "ainvs": [ "1", "-1", "0", "-56880994", "168642718624" ], "rank_lower_bound": 11, "naive_height": 65.22507333308967, "faltings_height": 3.333050043452092, "conductor": "252948166615918192888894", "bad_primes": [ "2", "1601", "78996928986857649247" ], "discriminant": "-505896333231836385777788", "regulator": "91182620.3414038712920906292443019334784271018144625666456613", "points": [ [ "4432", "57680" ], [ "4198", "59750" ], [ "4568", "61852" ], [ "4056", "66140" ], [ "4606", "63654" ], [ "4018", "68352" ], [ "3986", "70350" ], [ "4654", "66278" ], [ "4688", "68352" ], [ "3830", "81508" ], [ "3738", "88910" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:48", "updated_at": "2026-07-01 22:48:35" }, { "id": 145, "curve_key": "4976284521:-350646573884085", "ainvs": [ "1", "-1", "0", "-103672594", "405866860144" ], "rank_lower_bound": 11, "naive_height": 66.98384810821862, "faltings_height": 3.3596863109018527, "conductor": "80075902593366056993306", "bad_primes": [ "2", "148492754093", "269628989921" ], "discriminant": "160151805186732113986612", "regulator": "7518277.47498296156525289407962005873548111778486054296264713", "points": [ [ "5578", "-36138" ], [ "6573", "-94810" ], [ "600", "-586712" ], [ "3011", "-349356" ], [ "6243", "-47063" ], [ "-2687", "-814150" ], [ "6323", "52612" ], [ "6190", "32702" ], [ "7357", "-206922" ], [ "5717", "-1827" ], [ "22643/4", "-176895/8" ] ], "submitter": "cocoxhuang", "commentary": null, "created_at": "2026-06-24 02:21:27", "updated_at": "2026-07-01 22:49:32" }, { "id": 33, "curve_key": "49613792119803779776:-349232792964630602024344083040", "ainvs": [ "0", "1", "0", "-1033620669162578745", "404204621142300751511020144" ], "rank_lower_bound": 11, "naive_height": 136.05240160770538, "faltings_height": 9.09640726507286, "conductor": "535701165729047061464726179383779596", "bad_primes": [ "2", "13", "17", "23", "29", "31", "37", "41", "43", "47", "53", "59", "3057053387069549573" ], "discriminant": "93873316108015714157059941602262270600638822314912592", "regulator": "16764246962.6599538425546112945285195822333923272100049593443", "points": [ [ "567020697", "651563123419" ], [ "572060549", "345977302867" ], [ "618262355", "1218864665837" ], [ "9588741225/16", "1394897103107/64" ], [ "2062532261/4", "23090550241369/8" ], [ "573619772", "207397662806" ], [ "553608987", "1286321677901" ], [ "647275079", "2520583821587" ], [ "573703341", "197837573149" ], [ "611109417", "878062026491" ], [ "23257761261641/38809", "77388276951103391/7645373" ] ], "submitter": "David Renshaw", "commentary": "Found by Schneiders - Zimmer (1991). A curve of rank exactly 11, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:12", "updated_at": "2026-07-01 22:49:32" }, { "id": 34, "curve_key": "5534713447956935783296:-263476812012507022000959221131360", "ainvs": [ "0", "1", "0", "-115306530165769495485", "304950013864929098742150007444" ], "rank_lower_bound": 11, "naive_height": 150.19598024032572, "faltings_height": 10.55322744350137, "conductor": "159117919909446340006437537659166966132164", "bad_primes": [ "2", "19", "23", "29", "31", "37", "47", "53", "59", "61", "67", "79", "83", "97", "7163230956091032353" ], "discriminant": "57942789150218118922071208483979411230699716000862987088178512", "regulator": "251615995233.483808978493583609207799224828706024666003168936", "points": [ [ "16720669911", "1746918971478919" ], [ "-27494384469/25", "82009061233472771/125" ], [ "24003278596/9", "3457124546861636/27" ], [ "-473203304125/49", "246826926055649591/343" ], [ "143049082761/64", "123719807142058949/512" ], [ "-5963359769", "883457619817881" ], [ "6678087633629/676", "6335744808950599147/17576" ], [ "10500034749/4", "1141436027393173/8" ], [ "9063959795", "66852115553887" ], [ "70954716481/25", "2933723417746929/125" ], [ "13823082740", "1162901547372488" ] ], "submitter": "David Renshaw", "commentary": "Found by Schneiders - Zimmer (1991). A curve of rank exactly 11, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:12", "updated_at": "2026-07-01 22:49:32" }, { "id": 210, "curve_key": "704545914222730715359034280289:-515073146828791295390840612344745256877902737", "ainvs": [ "1", "1", "1", "-14678039879640223236646547506", "596149475496280105667226784565254703843519" ], "rank_lower_bound": 11, "naive_height": 206.18205303849962, "faltings_height": 15.168483543162193, "conductor": "44702660536826547810138278991954523939494764170080309889431835489890", "bad_primes": [ "2", "3", "5", "7", "13", "17", "19", "157", "2004720611", "630581423473", "255430951802014070304557438075659110521" ], "discriminant": "48857429671295622571237789586614535277793367429869518495768693667778413208750462668800", "regulator": "2751784961168.25749229920921519575526880971751709069524886858", "points": [ [ "58446067465881769", "-14129669742311093189766125" ], [ "59805178857601951/9", "-14622998659211562714239875/27" ], [ "41268971140291161/64", "-8243875117670686839240415/512" ], [ "308108574052911809/841", "-162496149169433149662813345/24389" ], [ "2136555534033197/16", "-64549605615325939384071/64" ], [ "2447918870143671799281/27321529", "2428644462538774240003361334025/142809632083" ], [ "91033697994269", "119840245127165271375" ], [ "3579092917301768024617/34539129", "87981230877416815421786034759967/202986461133" ], [ "8040116944201663249569/45010681", "578803785815401822474435058544295/301976658829" ], [ "324903547043833", "5488603739649485397427" ], [ "360854332952117", "6502958279889840406551" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥11. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=15681/13: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 11 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:04:34", "updated_at": "2026-07-01 22:49:32" }, { "id": 212, "curve_key": "10634696675535523446376517660944:-30573625125649705614695987633207721066059380672", "ainvs": [ "0", "1", "0", "-221556180740323405132844117936", "35386140191724122461245294467670188433973860" ], "rank_lower_bound": 11, "naive_height": 214.32502415017666, "faltings_height": 15.842747841238426, "conductor": "1103561624055499058867562340698878392772504928025988266715523317532246643920", "bad_primes": [ "2", "3", "5", "11", "31", "157", "670606297099", "1575838430456954508271967", "81274068710384465721193186106423" ], "discriminant": "155094517685876658621636759147912906994801674312562010974276152381499783783841112416793600", "regulator": "15079211501874.5923514849165940580973990610604958433553929196", "points": [ [ "1439725582641991678/9", "-1727498601505665033867479240/27" ], [ "1609812018760123810/81", "-2041934892838682216734870460/729" ], [ "382889599831597753/144", "-233409324592028354395076875/1728" ], [ "1461300830275451952/841", "-1706610970573605828131905230/24389" ], [ "3003181201931493/4", "-136765210570118860311615/8" ], [ "4642447936168800893203/13344409", "31359710078558692581991890119040/48747126077" ], [ "24321114180952583584546/63696361", "1291845898306026112352226583813164/508360657141" ], [ "38661152893172916450454/78234025", "4722711530104732719979077042193992/691979951125" ], [ "721344004004635", "15840139419502659908460" ], [ "779580898158616", "18342645394695663676806" ], [ "16344491930375377/16", "-1893202461186603241978305/64" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥11. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=14612/9: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 11 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:04:44", "updated_at": "2026-07-01 22:49:33" }, { "id": 211, "curve_key": "103895420219460887976030758560849:-1059351374675894073952680990935796750743854909657", "ainvs": [ "1", "0", "0", "-2164487921238768499500640803351", "1226101128097099442256424377019056466085320681" ], "rank_lower_bound": 11, "naive_height": 221.1634825477226, "faltings_height": 16.165246271644328, "conductor": "76884160347970864077673900010863037738725767115232422182209303915570", "bad_primes": [ "2", "5", "7", "13", "17", "23", "31", "157", "33196799", "2096948303", "1944251946437", "25233503438885314304619774163" ], "discriminant": "-434796625061430848546124049916980604145395749660512866802185034140618458039551190139699200", "regulator": "6574796646890.56238082668096592613712251120723802057237124620", "points": [ [ "193471633301052616639334/1352569", "-85094808570897910903821838655812881/1573037747" ], [ "405207134144905894/9", "-257802161495288662231116521/27" ], [ "550091167141970389100/130321", "-12204497223776414581241469614979/47045881" ], [ "209300708887435110/361", "88389056748088433517564679/6859" ], [ "793448046452518", "2866253390847396035317" ], [ "846049037447238", "662750304475737010357" ], [ "7900678100479420/9", "42649563397126798370693/27" ], [ "1030036895154710", "9457437854839529239509" ], [ "1710543209524774", "50285471431872021894373" ], [ "2517886218947724", "108346380885156670615183" ], [ "51824229593028927046/15625", "341289027873148644301074534281/1953125" ] ], "submitter": "Seewoo Lee", "commentary": "Rank ≥11. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=2326/23: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 11 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.", "created_at": "2026-06-25 16:04:39", "updated_at": "2026-07-01 22:49:33" }, { "id": 60, "curve_key": "73759881:-559794203685", "ainvs": [ "1", "-1", "0", "-1536664", "648294124" ], "rank_lower_bound": 10, "naive_height": 54.34897657161224, "faltings_height": 2.5105052071053513, "conductor": "25440555737235843986", "bad_primes": [ "2", "53993", "235591240876001" ], "discriminant": "50881111474471687972", "regulator": "488600.0685314399839133782564429436509012776371221864320710", "points": [ [ "-1412", "1890" ], [ "-1411", "3123" ], [ "-1386", "11378" ], [ "-1372", "13822" ], [ "-1353", "16463" ], [ "-1350", "16832" ], [ "-1329", "19154" ], [ "-1274", "23810" ], [ "-1259", "24840" ], [ "-1237", "26215" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Elkies-Watkins, \"Elliptic curves of large rank and small conductor\" (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=10: [1,-1,0,-1536664,648294124], |Delta|=50881111474471687972; their I-column records 207 integral x-coordinates. Also appears in Table 2 with conductor N=25440555737235843986 and |Delta|/N=2. The submitted points certify rank >= 10 here.", "created_at": "2026-05-29 00:04:02", "updated_at": "2026-07-01 22:49:33" }, { "id": 82, "curve_key": "85863216:-728508095640", "ainvs": [ "0", "0", "1", "-1788817", "843180666" ], "rank_lower_bound": 10, "naive_height": 54.8047982291504, "faltings_height": 2.5323775332354264, "conductor": "59202439687694448757", "bad_primes": [ "59202439687694448757" ], "discriminant": "59202439687694448757", "regulator": "774709.5755443039950812393750908533891851784680859970994988", "points": [ [ "-32", "30006" ], [ "38", "27843" ], [ "-84", "31509" ], [ "-87", "31593" ], [ "105", "25622" ], [ "214", "21683" ], [ "-234", "35340" ], [ "-291", "36593" ], [ "313", "17718" ], [ "355", "15902" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Elkies-Watkins, Elliptic curves of large rank and small conductor (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=10: [0,0,1,-1788817,843180666], |Delta|=59202439687694448757, I=176. The submitted integral points give the rank >= 10 witness checked by the site.", "created_at": "2026-06-02 18:44:34", "updated_at": "2026-07-01 22:50:30" }, { "id": 83, "curve_key": "89112016:-927152864920", "ainvs": [ "0", "1", "1", "-1856500", "1072474760" ], "rank_lower_bound": 10, "naive_height": 55.11076858348722, "faltings_height": 2.561371787165621, "conductor": "87950374485438204043", "bad_primes": [ "1651819847", "53244531869" ], "discriminant": "-87950374485438204043", "regulator": "995214.3440764446000421615341619391028494443800016433750965", "points": [ [ "26", "32003" ], [ "-80", "34935" ], [ "91", "30071" ], [ "116", "29303" ], [ "211", "26271" ], [ "-244", "38871" ], [ "248", "25047" ], [ "349", "21614" ], [ "-433", "42371" ], [ "-463", "42813" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Elkies-Watkins, Elliptic curves of large rank and small conductor (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=10: [0,1,1,-1856500,1072474760], |Delta|=87950374485438204043, I=154. The submitted integral points give the rank >= 10 witness checked by the site.", "created_at": "2026-06-02 18:44:35", "updated_at": "2026-07-01 22:50:31" }, { "id": 84, "curve_key": "117049296:-1334964971160", "ainvs": [ "0", "0", "1", "-2438527", "1545098346" ], "rank_lower_bound": 10, "naive_height": 55.83985233721011, "faltings_height": 2.5934334409263733, "conductor": "103294665688000244363", "bad_primes": [ "568841213", "181587872551" ], "discriminant": "-103294665688000244363", "regulator": "883114.6071087112093983008109847701591149707946289746317701", "points": [ [ "-17", "39831" ], [ "20", "38682" ], [ "57", "37500" ], [ "67", "37175" ], [ "-127", "43043" ], [ "169", "33731" ], [ "281", "29699" ], [ "-324", "47970" ], [ "-353", "48599" ], [ "354", "26948" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Elkies-Watkins, Elliptic curves of large rank and small conductor (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=10: [0,0,1,-2438527,1545098346], |Delta|=103294665688000244363, I=173. The submitted integral points give the rank >= 10 witness checked by the site.", "created_at": "2026-06-02 18:44:35", "updated_at": "2026-07-01 22:50:31" }, { "id": 101, "curve_key": "216650217:-3210180689781", "ainvs": [ "1", "-1", "0", "-4513546", "3716615296" ], "rank_lower_bound": 10, "naive_height": 57.59469668227534, "faltings_height": 2.6470753815544263, "conductor": "39432942782223365758", "bad_primes": [ "2", "73523", "268167395115973" ], "discriminant": "-78865885564446731516", "regulator": "626053.0026059040614370288739650539139511584443021140154468", "points": [ [ "1216", "4400" ], [ "1250", "4536" ], [ "1254", "4606" ], [ "1174", "5310" ], [ "1286", "5500" ], [ "1154", "6040" ], [ "1312", "6552" ], [ "1320", "6916" ], [ "1434", "13120" ], [ "942", "16852" ] ], "submitter": "David Renshaw", "commentary": "EW 2004 (Elkies-Watkins, ANTS-VI, arXiv:math/0403374) Table 2 low-conductor rank-10 list; this curve was absent from the leaderboard. Rank-10 witness via 10 independent rational points (Neron-Tate pairing).", "created_at": "2026-06-23 23:50:32", "updated_at": "2026-07-01 22:50:31" }, { "id": 95, "curve_key": "459908176:-9685738450880", "ainvs": [ "0", "-1", "0", "-9581420", "11213539236" ], "rank_lower_bound": 10, "naive_height": 59.83961223038972, "faltings_height": 2.8778622999438346, "conductor": "20882718586294952952", "bad_primes": [ "2", "3", "870113274428956373" ], "discriminant": "2004740984284315483392", "regulator": "306050.3022149287911853503237839069713678792795916714160689", "points": [ [ "42653", "8786262" ], [ "4681133/4", "-10128025065/8" ], [ "10412633/5041", "5441633328/357911" ], [ "1469", "-17502" ], [ "43673/16", "4698507/64" ], [ "-2784935/784", "-446425237/21952" ], [ "5784418/841", "12688027012/24389" ], [ "18425", "2467644" ], [ "819644/625", "469248978/15625" ], [ "1980", "-894" ] ], "submitter": "Andrew Sutherland", "commentary": null, "created_at": "2026-06-23 13:29:11", "updated_at": "2026-07-01 22:50:32" }, { "id": 99, "curve_key": "489317241:-10925357623245", "ainvs": [ "1", "-1", "0", "-10194109", "12647638369" ], "rank_lower_bound": 10, "naive_height": 60.04421518153516, "faltings_height": 2.86548777400581, "conductor": "22006161865320788846", "bad_primes": [ "2", "29", "272446637", "1392627151" ], "discriminant": "-1276357388188605753068", "regulator": "303375.5907842359104425786052982343182424858155554328186436", "points": [ [ "1740", "12383" ], [ "1566", "22069" ], [ "1392", "33263" ], [ "1334", "37033" ], [ "1943", "12180" ], [ "1653", "16820" ], [ "2784", "75023" ], [ "2233", "30740" ], [ "3190", "110577" ], [ "1015", "57333" ] ], "submitter": "David Renshaw", "commentary": "EW 2004 (Elkies-Watkins, ANTS-VI, arXiv:math/0403374) Table 2 low-conductor rank-10 list; this curve was absent from the leaderboard. Rank-10 witness via 10 independent rational points (Neron-Tate pairing).", "created_at": "2026-06-23 23:50:29", "updated_at": "2026-07-01 22:50:32" }, { "id": 49, "curve_key": "782994576:-22438220526360", "ainvs": [ "0", "0", "1", "-16312387", "25970162646" ], "rank_lower_bound": 10, "naive_height": 61.48357378851376, "faltings_height": 3.0269363701664456, "conductor": "10189285026863130793", "bad_primes": [ "11", "1297", "4423", "75707", "2132839" ], "discriminant": "-13561938370754827085483", "regulator": "990834.3392128087320646545893821685852413506370864254350345", "points": [ [ "-4674", "10235" ], [ "-4036", "161441" ], [ "-2413", "226455" ], [ "-305", "175832" ], [ "1277", "84980" ], [ "-4509", "88599" ], [ "2463", "27096" ], [ "3246", "84980" ], [ "4687", "229080" ], [ "7993", "637371" ] ], "submitter": "David Renshaw", "commentary": "Smallest known conductor of an elliptic curve with rank ≥ 10 (Elkies–Watkins, 2004). From their paper \"Elliptic Curves of Large Rank and Small Conductor\" (arXiv:math/0403374).", "created_at": "2026-05-28 18:40:50", "updated_at": "2026-07-01 22:50:32" }, { "id": 100, "curve_key": "1011790416:-30835288039320", "ainvs": [ "0", "0", "1", "-21078967", "35688990786" ], "rank_lower_bound": 10, "naive_height": 62.2049618626486, "faltings_height": 3.1146519240176422, "conductor": "22630148490190627609", "bad_primes": [ "41", "53", "10414242287248333" ], "discriminant": "49175312669184233794357", "regulator": "283148.7940006095034539845739385244675957106444863463613555", "points": [ [ "2190", "5432" ], [ "17", "187964" ], [ "-4329", "214040" ], [ "2137", "20060" ], [ "3091", "8135" ], [ "3144", "22233" ], [ "1872", "52814" ], [ "1819", "58008" ], [ "1944", "45366" ], [ "3215", "33927" ] ], "submitter": "David Renshaw", "commentary": "EW 2004 (Elkies-Watkins, ANTS-VI, arXiv:math/0403374) Table 2 low-conductor rank-10 list; this curve was absent from the leaderboard. Rank-10 witness via 10 independent rational points (Neron-Tate pairing).", "created_at": "2026-06-23 23:50:30", "updated_at": "2026-07-01 22:51:29" }, { "id": 31, "curve_key": "98918170715321792161:-896711378899860072200428904305", "ainvs": [ "1", "0", "0", "-2060795223235870670", "1037860392073475481628988676" ], "rank_lower_bound": 10, "naive_height": 138.12247387142565, "faltings_height": 9.477932235718134, "conductor": "240767746179482941231124244768162", "bad_primes": [ "2", "3", "17", "19", "23", "29", "31", "37", "41", "43", "47", "53", "697675341112349" ], "discriminant": "94793787423403087849353656317664770631480997068191076352", "regulator": "17736686.04120010720386049179366267873878228181167867826206", "points": [ [ "150913924", "27023891033530" ], [ "1309368796", "24172933593538" ], [ "1714332274", "50430125403496" ], [ "1088122156", "9154278492706" ], [ "1080521116", "8523778305538" ], [ "1048497604", "5456990794690" ], [ "153591884988", "60191191934061474" ], [ "1028141818", "2427298113664" ], [ "1023547336", "925461825238" ], [ "-1104208292", "-44351144023742" ] ], "submitter": "David Renshaw", "commentary": "Found by Kretschmer (1986). A curve of rank exactly 10, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:10", "updated_at": "2026-07-01 22:51:29" }, { "id": 32, "curve_key": "163917537878828567281:-1263721367114133797432399098825", "ainvs": [ "1", "0", "0", "-3414948705808928485", "1462640470912335058581199361" ], "rank_lower_bound": 10, "naive_height": 139.6376854723492, "faltings_height": 9.677832038105548, "conductor": "929053397624034727388941088597323782", "bad_primes": [ "2", "3", "17", "19", "23", "29", "31", "37", "43", "47", "53", "59", "281", "74177", "4756942171609" ], "discriminant": "1624596034552120480493303406632087249542571198120190723072", "regulator": "347100868.2553289242017767590904381615260046941546101545684", "points": [ [ "1651118138", "18038584222223" ], [ "1588614902", "6839177749439" ], [ "-1189653814", "-61979712512809" ], [ "-700495294", "-59253910115977" ], [ "-147325510", "-44300624514889" ], [ "11898974402", "1282784596947959" ], [ "448355771899346/126025", "8290865418217263800429/44738875" ], [ "-621829360", "-57841818977839" ], [ "1692223027970/841", "1276149459553975651/24389" ], [ "-380499314", "-52028047503377" ] ], "submitter": "David Renshaw", "commentary": "Found by Kretschmer (1986). A curve of rank exactly 10, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:11", "updated_at": "2026-07-01 22:51:30" }, { "id": 59, "curve_key": "8036112:-26145789336", "ainvs": [ "0", "0", "1", "-167419", "30261330" ], "rank_lower_bound": 9, "naive_height": 47.97390798967291, "faltings_height": 1.9827068050306047, "conductor": "95276302704064331", "bad_primes": [ "95276302704064331" ], "discriminant": "-95276302704064331", "regulator": "294208.68480812036499615812853149590022566024193525846092", "points": [ [ "-480", "174" ], [ "-479", "743" ], [ "-478", "1035" ], [ "-477", "1260" ], [ "-463", "2919" ], [ "-439", "4376" ], [ "-422", "5075" ], [ "-384", "6158" ], [ "-347", "6824" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Elkies-Watkins, \"Elliptic curves of large rank and small conductor\" (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=9: [0,0,1,-167419,30261330], |Delta|=95276302704064331; their I-column records 135 integral x-coordinates. The submitted points certify rank >= 9 here.", "created_at": "2026-05-29 00:04:01", "updated_at": "2026-07-01 22:51:30" }, { "id": 48, "curve_key": "6480201:-83909859525", "ainvs": [ "1", "-1", "0", "-135004", "97151644" ], "rank_lower_bound": 9, "naive_height": 50.30601791742373, "faltings_height": 2.2491384627836597, "conductor": "32107342006814614", "bad_primes": [ "2", "61", "263174934482087" ], "discriminant": "-3917095724831382908", "regulator": "33288.740573083599055593661213206843477412693238540465426", "points": [ [ "-556", "-42" ], [ "-452", "-7878" ], [ "-233", "-10650" ], [ "-25", "-10013" ], [ "197", "8743" ], [ "446", "10980" ], [ "801", "22015" ], [ "1219", "41138" ], [ "1876", "79334" ] ], "submitter": "David Renshaw", "commentary": "Smallest known conductor of an elliptic curve with rank ≥ 9 (Elkies–Watkins, 2004). From their paper \"Elliptic Curves of Large Rank and Small Conductor\" (arXiv:math/0403374).", "created_at": "2026-05-28 18:27:55", "updated_at": "2026-07-01 22:51:30" }, { "id": 80, "curve_key": "19303536:-83195714520", "ainvs": [ "0", "0", "1", "-402157", "96291336" ], "rank_lower_bound": 9, "naive_height": 50.327396548587, "faltings_height": 2.087850183987795, "conductor": "157107745029925477", "bad_primes": [ "157107745029925477" ], "discriminant": "157107745029925477", "regulator": "1253079.4942741755784427259681451604931159440062484019892", "points": [ [ "-9", "9995" ], [ "16", "9479" ], [ "46", "8825" ], [ "61", "8484" ], [ "74", "8181" ], [ "-76", "11243" ], [ "102", "7505" ], [ "108", "7356" ], [ "-113", "11844" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Elkies-Watkins, Elliptic curves of large rank and small conductor (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=9: [0,0,1,-402157,96291336], |Delta|=157107745029925477, I=131. The submitted integral points give the rank >= 9 witness checked by the site.", "created_at": "2026-06-02 18:44:32", "updated_at": "2026-07-01 22:51:30" }, { "id": 79, "curve_key": "24696336:-121657002840", "ainvs": [ "0", "0", "1", "-514507", "140806716" ], "rank_lower_bound": 9, "naive_height": 51.06649635154127, "faltings_height": 2.114888621874499, "conductor": "151673348057775877", "bad_primes": [ "409", "6129953", "60496301" ], "discriminant": "151673348057775877", "regulator": "82462.036760633052867653702337648351572050396724991183500", "points": [ [ "8", "11691" ], [ "-22", "12333" ], [ "-48", "12860" ], [ "83", "9933" ], [ "-114", "14070" ], [ "159", "7938" ], [ "184", "7236" ], [ "198", "6833" ], [ "-207", "15441" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Elkies-Watkins, Elliptic curves of large rank and small conductor (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=9: [0,0,1,-514507,140806716], |Delta|=151673348057775877, I=126. The submitted integral points give the rank >= 9 witness checked by the site.", "created_at": "2026-06-02 18:44:31", "updated_at": "2026-07-01 22:51:31" }, { "id": 132, "curve_key": "29427321:-85304293965", "ainvs": [ "1", "-1", "0", "-613069", "98885089" ], "rank_lower_bound": 9, "naive_height": 51.59230225951362, "faltings_height": 2.347437883951686, "conductor": "43537345103385386", "bad_primes": [ "2", "11", "179906384724733" ], "discriminant": "10536037515019263412", "regulator": "28128.954293126579199618116289534715658074636283516753699", "points": [ [ "168", "697" ], [ "102", "6065" ], [ "36", "8749" ], [ "-52", "11455" ], [ "-184", "14425" ], [ "-228", "15173" ], [ "740", "6703" ], [ "806", "10905" ], [ "157", "2468" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:39", "updated_at": "2026-07-01 22:52:28" }, { "id": 156, "curve_key": "36474249:-122606032293", "ainvs": [ "1", "-1", "0", "-759880", "142095100" ], "rank_lower_bound": 9, "naive_height": 52.236351188149, "faltings_height": 2.3990696156237847, "conductor": "1938196651043119330", "bad_primes": [ "2", "5", "637771", "303901659223" ], "discriminant": "19381966510431193300", "regulator": "100835.16934808000041574528314722060871634978015464820244", "points": [ [ "11", "11559" ], [ "-37", "-13026" ], [ "38", "10624" ], [ "-50", "-13390" ], [ "-103", "-14756" ], [ "110", "7680" ], [ "130", "6680" ], [ "135", "6410" ], [ "147", "5719" ] ], "submitter": "Edgar Costa", "commentary": "Found by the ec-smallcond GPU integral-point search (Elkies-Watkins style large-rank/small-conductor sweep); rank >= 9 certified from 9 independent points via PARI/GP ellrank (returned rank exactly 9).", "created_at": "2026-06-24 21:09:14", "updated_at": "2026-07-01 22:52:28" }, { "id": 81, "curve_key": "39677232:-250522113816", "ainvs": [ "0", "0", "1", "-826609", "289956150" ], "rank_lower_bound": 9, "naive_height": 52.493626064563166, "faltings_height": 2.1805172142484204, "conductor": "172539371946838571", "bad_primes": [ "69238607", "2491953253" ], "discriminant": "-172539371946838571", "regulator": "91567.683284080923712615686709609971136528575064325022194", "points": [ [ "17", "16610" ], [ "-57", "18354" ], [ "60", "15510" ], [ "103", "14349" ], [ "-143", "20130" ], [ "-188", "20945" ], [ "243", "10170" ], [ "248", "10010" ], [ "331", "7253" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Elkies-Watkins, Elliptic curves of large rank and small conductor (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=9: [0,0,1,-826609,289956150], |Delta|=172539371946838571, I=120. The submitted integral points give the rank >= 9 witness checked by the site.", "created_at": "2026-06-02 18:44:33", "updated_at": "2026-07-01 22:52:28" }, { "id": 135, "curve_key": "51656857:-428648563261", "ainvs": [ "1", "0", "1", "-1076185", "496031340" ], "rank_lower_bound": 9, "naive_height": 53.5678064405781, "faltings_height": 2.4507827662879853, "conductor": "77211251506212554", "bad_primes": [ "2", "43", "897805250072239" ], "discriminant": "-26560670518137118576", "regulator": "53538.096568535612150394273481249423219180410009474496562", "points": [ [ "679", "8518" ], [ "335", "12990" ], [ "1023", "21074" ], [ "-525", "30534" ], [ "-1041", "22622" ], [ "-1213", "4734" ], [ "120", "19139" ], [ "34", "21418" ], [ "1668", "56979" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:43", "updated_at": "2026-07-01 22:52:28" }, { "id": 155, "curve_key": "57058041:-427399227405", "ainvs": [ "1", "-1", "0", "-1188709", "494972209" ], "rank_lower_bound": 9, "naive_height": 53.57873871263536, "faltings_height": 2.322290109872686, "conductor": "893866404426681866", "bad_primes": [ "2", "347", "34147", "37719036437" ], "discriminant": "1787732808853363732", "regulator": "416895.39959299299883511375405071915406693053311770994384", "points": [ [ "48", "20905" ], [ "65", "20412" ], [ "212", "15783" ], [ "-228", "-27347" ], [ "260", "14133" ], [ "-260", "-27913" ], [ "-293", "-28455" ], [ "329", "11643" ], [ "404", "8773" ] ], "submitter": "Edgar Costa", "commentary": "Found by the ec-smallcond GPU integral-point search (Elkies-Watkins style large-rank/small-conductor sweep); rank >= 9 certified from 9 independent points via PARI/GP ellrank (returned rank exactly 9).", "created_at": "2026-06-24 21:09:12", "updated_at": "2026-07-01 22:52:29" }, { "id": 153, "curve_key": "59546361:-450133503885", "ainvs": [ "1", "-1", "0", "-1240549", "521297989" ], "rank_lower_bound": 9, "naive_height": 53.706797230758156, "faltings_height": 2.3725445185657126, "conductor": "616136865466771394", "bad_primes": [ "2", "83", "14549", "255114981391" ], "discriminant": "4929094923734171152", "regulator": "102053.46086070287983607086017344691046650256242692048241", "points": [ [ "-10", "-23097" ], [ "62", "21055" ], [ "73", "20727" ], [ "183", "17240" ], [ "195", "16837" ], [ "210", "16327" ], [ "278", "13927" ], [ "-286", "-29057" ], [ "-410", "-30793" ] ], "submitter": "Edgar Costa", "commentary": "Found by the ec-smallcond GPU integral-point search (Elkies-Watkins style large-rank/small-conductor sweep); rank >= 9 certified from 9 independent points via PARI/GP ellrank (returned rank exactly 9).", "created_at": "2026-06-24 21:09:07", "updated_at": "2026-07-01 22:52:29" }, { "id": 134, "curve_key": "71665321:-606480135445", "ainvs": [ "1", "0", "1", "-1493028", "701820182" ], "rank_lower_bound": 9, "naive_height": 54.26255256151007, "faltings_height": 2.25720433224642, "conductor": "72070075910145406", "bad_primes": [ "2", "1187", "30358077468469" ], "discriminant": "144140151820290812", "regulator": "58361.246530795783731326957120654161428495711241018202990", "points": [ [ "719", "37" ], [ "729", "613" ], [ "679", "763" ], [ "731", "715" ], [ "675", "961" ], [ "665", "1441" ], [ "745", "1401" ], [ "659", "1723" ], [ "688", "285" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:42", "updated_at": "2026-07-01 22:52:29" }, { "id": 152, "curve_key": "71803641:-601907338125", "ainvs": [ "1", "-1", "0", "-1495909", "697025989" ], "rank_lower_bound": 9, "naive_height": 54.26833722911741, "faltings_height": 2.3906563519740556, "conductor": "572200654902875954", "bad_primes": [ "2", "286100327451437977" ], "discriminant": "4577605239223007632", "regulator": "102844.25924766478156295600134311296324227256181616231024", "points": [ [ "19", "25848" ], [ "-106", "-29177" ], [ "-122", "-29565" ], [ "161", "21375" ], [ "187", "20493" ], [ "273", "17440" ], [ "-317", "-33595" ], [ "-354", "-34205" ], [ "405", "12347" ] ], "submitter": "Edgar Costa", "commentary": "Found by the ec-smallcond GPU integral-point search (Elkies-Watkins style large-rank/small-conductor sweep); rank >= 9 certified from 9 independent points via PARI/GP ellrank (returned rank exactly 9).", "created_at": "2026-06-24 21:09:05", "updated_at": "2026-07-01 22:53:26" }, { "id": 150, "curve_key": "83680761:-765901311885", "ainvs": [ "1", "-1", "0", "-1743349", "886895689" ], "rank_lower_bound": 9, "naive_height": 54.728638325455414, "faltings_height": 2.3120879913684127, "conductor": "183107934630169474", "bad_primes": [ "2", "244254379", "374830403" ], "discriminant": "-366215869260338948", "regulator": "293591.30673220840727887138511462027952831615032133636905", "points": [ [ "-28", "-30575" ], [ "56", "28069" ], [ "-87", "-32173" ], [ "130", "25673" ], [ "-159", "-33980" ], [ "176", "24109" ], [ "264", "20963" ], [ "-285", "-36743" ], [ "-302", "-37075" ] ], "submitter": "Edgar Costa", "commentary": "Found by the ec-smallcond GPU integral-point search (Elkies-Watkins style large-rank/small-conductor sweep); rank >= 9 certified from 9 independent points via PARI/GP ellrank (returned rank exactly 9).", "created_at": "2026-06-24 21:06:39", "updated_at": "2026-07-01 22:53:26" }, { "id": 144, "curve_key": "95837616:-886251581280", "ainvs": [ "0", "0", "0", "-1996617", "1025754145" ], "rank_lower_bound": 9, "naive_height": 55.134497451692134, "faltings_height": 2.539020430194318, "conductor": "56448695244116556", "bad_primes": [ "2", "3", "1051", "1303", "1144996807" ], "discriminant": "54868131777281292432", "regulator": "32469.293473600010273681102961273115152708194679048581104", "points": [ [ "653", "-639" ], [ "971", "1593" ], [ "1103", "12861" ], [ "176", "-26073" ], [ "629", "4329" ], [ "1157", "-16263" ], [ "1439", "33651" ], [ "536", "-10467" ], [ "-661", "45351" ] ], "submitter": "cocoxhuang", "commentary": null, "created_at": "2026-06-24 02:21:09", "updated_at": "2026-07-01 22:53:27" }, { "id": 133, "curve_key": "184119312:-2496865590936", "ainvs": [ "0", "0", "1", "-3835819", "2889890730" ], "rank_lower_bound": 9, "naive_height": 57.093283620638616, "faltings_height": 2.5119487290750904, "conductor": "62986816173592807", "bad_primes": [ "67", "940101733934221" ], "discriminant": "4220116683630718069", "regulator": "54040.743340650066495609885765423138876129719521622311100", "points": [ [ "1108", "234" ], [ "1175", "2244" ], [ "1041", "4991" ], [ "773", "19664" ], [ "572", "29714" ], [ "1711", "36548" ], [ "237", "44655" ], [ "2180", "69914" ], [ "2247", "74939" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:40", "updated_at": "2026-07-01 22:54:24" }, { "id": 154, "curve_key": "188474832:-2655482353344", "ainvs": [ "0", "0", "0", "-3926559", "3073474946" ], "rank_lower_bound": 9, "naive_height": 57.21529486292699, "faltings_height": 2.675101772963943, "conductor": "716225953128355512", "bad_primes": [ "2", "3", "1663", "5669", "5801", "60631" ], "discriminant": "-206273074500966387456", "regulator": "61186.124504384347188884237562748022432111958022609102918", "points": [ [ "-26", "-56352" ], [ "67", "-53016" ], [ "-202", "-62116" ], [ "263", "-45376" ], [ "265", "-45294" ], [ "397", "-39714" ], [ "509", "-34738" ], [ "-551", "-71202" ], [ "-563", "-71454" ] ], "submitter": "Edgar Costa", "commentary": "Found by the ec-smallcond GPU integral-point search (Elkies-Watkins style large-rank/small-conductor sweep); rank >= 9 certified from 9 independent points via PARI/GP ellrank (returned rank exactly 9).", "created_at": "2026-06-24 21:09:10", "updated_at": "2026-07-01 22:54:24" }, { "id": 151, "curve_key": "256661184:-4096122507648", "ainvs": [ "0", "0", "0", "-5347108", "4740882532" ], "rank_lower_bound": 9, "naive_height": 58.08980226863035, "faltings_height": 2.6658136587624983, "conductor": "467760769046419720", "bad_primes": [ "2", "5", "131", "62119", "1437037337" ], "discriminant": "74841723047427155200", "regulator": "53110.452790351633137598676208541784979819556057572982627", "points": [ [ "16", "-68230" ], [ "72", "-66002" ], [ "-154", "-74570" ], [ "266", "-57770" ], [ "-296", "-79358" ], [ "321", "-55295" ], [ "488", "-47410" ], [ "-519", "-85885" ], [ "534", "-45142" ] ], "submitter": "Edgar Costa", "commentary": "Found by the ec-smallcond GPU integral-point search (Elkies-Watkins style large-rank/small-conductor sweep); rank >= 9 certified from 9 independent points via PARI/GP ellrank (returned rank exactly 9).", "created_at": "2026-06-24 21:09:02", "updated_at": "2026-07-01 22:54:24" }, { "id": 162, "curve_key": "49433743504:-13749204393788608", "ainvs": [ "0", "1", "0", "-1029869656", "15913087721444" ], "rank_lower_bound": 9, "naive_height": 74.31951471015851, "faltings_height": 4.1996249592052965, "conductor": "1312277573656056405360", "bad_primes": [ "2", "3", "5", "43", "599", "115831", "1832712667" ], "discriminant": "-39490526497340544133267123200", "regulator": "1725985.6630133732536627998069706320686406126996309353094", "points": [ [ "30800", "-3662262" ], [ "104384", "-32338710" ], [ "5201141/4", "-11858161785/8" ], [ "72821/4", "14306745/8" ], [ "122216/25", "414489186/125" ], [ "1528496/49", "1289868390/343" ], [ "1109966/25", "949593564/125" ], [ "3929870/169", "4680903876/2197" ], [ "10635650/529", "-22218226104/12167" ] ], "submitter": "Seewoo Lee", "commentary": "Use Fermigier-Mestre's quartic parametrization with (a_1, \\dots, a_6) = (0,6,12,14,15,23) and t = 1. This 6-tuple may not come from specialization of actual bivariate polynomial 6-tuples of Fermigier-Mestre.", "created_at": "2026-06-25 06:08:44", "updated_at": "2026-07-01 22:54:24" }, { "id": 30, "curve_key": "77191414209:-22079442640218273", "ainvs": [ "1", "-1", "1", "-1608154463", "25555312501831" ], "rank_lower_bound": 9, "naive_height": 75.26684674651447, "faltings_height": 4.184663345365095, "conductor": "9612838979769443646330", "bad_primes": [ "2", "3", "5", "75217", "1420015714498561" ], "discriminant": "-15946546326759934676023910400", "regulator": "254394.10864036720716591387454974845012295985529802681621", "points": [ [ "23045", "844952" ], [ "34381", "3285026" ], [ "25583", "1062950" ], [ "20803", "1040006" ], [ "21715", "923702" ], [ "17731", "1608326" ], [ "15371", "2106046" ], [ "19507", "1258166" ], [ "18715", "1409582" ] ], "submitter": "David Renshaw", "commentary": "Found by Brumer - Kramer (1977). A historical rank ≥ 9 record, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:09", "updated_at": "2026-07-01 22:54:25" }, { "id": 58, "curve_key": "789136:-1209159640", "ainvs": [ "0", "1", "1", "-16440", "1394010" ], "rank_lower_bound": 8, "naive_height": 41.826382885745815, "faltings_height": 1.524454279535257, "conductor": "561715239383323", "bad_primes": [ "23382011", "24023393" ], "discriminant": "-561715239383323", "regulator": "49900.6667347885574015353067843339561342699573634387980", "points": [ [ "-159", "116" ], [ "-151", "675" ], [ "-140", "985" ], [ "-138", "1026" ], [ "-124", "1241" ], [ "-114", "1341" ], [ "-93", "1458" ], [ "-91", "1464" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Elkies-Watkins, \"Elliptic curves of large rank and small conductor\" (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=8: [0,1,1,-16440,1394010], |Delta|=561715239383323; their I-column records 84 integral x-coordinates. The submitted points certify rank >= 8 here.", "created_at": "2026-05-29 00:04:00", "updated_at": "2026-07-01 22:54:25" }, { "id": 76, "curve_key": "1139376:-829756440", "ainvs": [ "0", "0", "1", "-23737", "960366" ], "rank_lower_bound": 8, "naive_height": 41.83797390626653, "faltings_height": 1.5176079485000202, "conductor": "457532830151317", "bad_primes": [ "457532830151317" ], "discriminant": "457532830151317", "regulator": "10423.4798881981483561708641513672204334594588544763000", "points": [ [ "5", "917" ], [ "-13", "1125" ], [ "17", "749" ], [ "-18", "1175" ], [ "-29", "1274" ], [ "33", "461" ], [ "36", "390" ], [ "38", "336" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Elkies-Watkins, \"Elliptic curves of large rank and small conductor\" (ANTS VI / arXiv:math/0403374), Table 4 low absolute-discriminant list for r=8: [0,0,1,-23737,960366], |Delta|=457532830151317; their I-column records 96 integral x-coordinates. The submitted integral points certify rank >= 8 here.", "created_at": "2026-06-01 15:47:39", "updated_at": "2026-07-01 22:54:25" }, { "id": 75, "curve_key": "1144624:-890361496", "ainvs": [ "0", "1", "1", "-23846", "1022562" ], "rank_lower_bound": 8, "naive_height": 41.85176027028323, "faltings_height": 1.5115276305530951, "conductor": "409086620841461", "bad_primes": [ "409086620841461" ], "discriminant": "409086620841461", "regulator": "13497.3934756072489297583974277373235085998065403862966", "points": [ [ "-5", "1068" ], [ "-7", "1090" ], [ "-15", "1173" ], [ "22", "713" ], [ "28", "614" ], [ "37", "438" ], [ "40", "366" ], [ "47", "88" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Elkies-Watkins, \"Elliptic curves of large rank and small conductor\" (ANTS VI / arXiv:math/0403374), Table 4 low absolute-discriminant list for r=8: [0,1,1,-23846,1022562], |Delta|=409086620841461; their I-column records 78 integral x-coordinates. The submitted integral points certify rank >= 8 here.", "created_at": "2026-06-01 15:47:38", "updated_at": "2026-07-01 22:55:23" }, { "id": 130, "curve_key": "3451161:-4755865725", "ainvs": [ "1", "-1", "0", "-71899", "5522449" ], "rank_lower_bound": 8, "naive_height": 45.162663762407796, "faltings_height": 1.7847536040199565, "conductor": "314658846776578", "bad_primes": [ "2", "17", "487", "19003433191" ], "discriminant": "10698400790403652", "regulator": "4338.70733787285461646054617796147519133767881422521590", "points": [ [ "80", "487" ], [ "216", "79" ], [ "12", "2153" ], [ "250", "1643" ], [ "63", "1082" ], [ "-260", "2697" ], [ "-294", "1235" ], [ "267", "2170" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:37", "updated_at": "2026-07-01 22:55:23" }, { "id": 47, "curve_key": "5106441:-11274515685", "ainvs": [ "1", "-1", "0", "-106384", "13075804" ], "rank_lower_bound": 8, "naive_height": 46.33803972609496, "faltings_height": 1.7639419667318916, "conductor": "249649566346838", "bad_primes": [ "2", "7", "17832111881917" ], "discriminant": "3495093928855732", "regulator": "4242.55153529952471732889342600472402806810042207796774", "points": [ [ "-375", "548" ], [ "-374", "854" ], [ "-370", "1482" ], [ "-1469/4", "14149/8" ], [ "-361", "2263" ], [ "-348", "2978" ], [ "-342", "3232" ], [ "-1337/4", "28119/8" ] ], "submitter": "David Renshaw", "commentary": "Smallest known conductor of an elliptic curve with rank ≥ 8 (Elkies–Watkins, 2004). From their paper \"Elliptic Curves of Large Rank and Small Conductor\" (arXiv:math/0403374).", "created_at": "2026-05-28 16:24:30", "updated_at": "2026-07-01 22:55:23" }, { "id": 131, "curve_key": "5966121:-12430981845", "ainvs": [ "1", "-1", "0", "-124294", "14418784" ], "rank_lower_bound": 8, "naive_height": 46.80482257634061, "faltings_height": 1.8936003389472096, "conductor": "315734078239402", "bad_primes": [ "2", "53", "1471", "2024896927" ], "discriminant": "33467812293376612", "regulator": "4378.89783766902967798504852401281835368190735016307202", "points": [ [ "78", "2240" ], [ "290", "1498" ], [ "131", "544" ], [ "-134", "5420" ], [ "25", "3353" ], [ "-240", "5632" ], [ "343", "3300" ], [ "502", "8600" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:38", "updated_at": "2026-07-01 22:55:24" }, { "id": 78, "curve_key": "9687081:-30131698005", "ainvs": [ "1", "-1", "0", "-201814", "34925104" ], "rank_lower_bound": 8, "naive_height": 48.258911100286376, "faltings_height": 1.7773692846159097, "conductor": "321754587851786", "bad_primes": [ "2", "2710699", "59349007" ], "discriminant": "643509175703572", "regulator": "6527.55172023458019453817713717187218224406628323898374", "points": [ [ "-5", "-5992" ], [ "-18", "-6200" ], [ "40", "5168" ], [ "62", "4728" ], [ "-69", "-6931" ], [ "79", "4373" ], [ "113", "3625" ], [ "154", "2658" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Elkies-Watkins, Elliptic curves of large rank and small conductor (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=8: [1,-1,0,-201814,34925104], |Delta|=643509175703572, I=109. The submitted integral points give the rank >= 8 witness checked by the site.", "created_at": "2026-06-02 18:44:31", "updated_at": "2026-07-01 22:55:24" }, { "id": 77, "curve_key": "10692057:-34976089629", "ainvs": [ "1", "-1", "0", "-222751", "40537273" ], "rank_lower_bound": 8, "naive_height": 48.555861023046056, "faltings_height": 1.78842300271875, "conductor": "292246301470558", "bad_primes": [ "2", "5904737", "24746767" ], "discriminant": "-584492602941116", "regulator": "30188.1405517264055310896306502872057316069977838122348", "points": [ [ "-21", "-6713" ], [ "26", "5883" ], [ "-37", "-6962" ], [ "-47", "-7111" ], [ "109", "4134" ], [ "116", "3973" ], [ "137", "3478" ], [ "194", "2047" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Elkies-Watkins, Elliptic curves of large rank and small conductor (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=8: [1,-1,0,-222751,40537273], |Delta|=584492602941116, I=101. The submitted integral points give the rank >= 8 witness checked by the site.", "created_at": "2026-06-02 18:44:30", "updated_at": "2026-07-01 22:55:24" }, { "id": 129, "curve_key": "23119824:-110775805632", "ainvs": [ "0", "0", "0", "-481663", "128212738" ], "rank_lower_bound": 8, "naive_height": 50.86860296784818, "faltings_height": 2.060643977084035, "conductor": "314214346667560", "bad_primes": [ "2", "5", "7855358666689" ], "discriminant": "50274295466809600", "regulator": "3808.43724670657225361488978860267064159808223292200143", "points": [ [ "369", "850" ], [ "429", "730" ], [ "379", "320" ], [ "449", "1570" ], [ "479", "2720" ], [ "249", "4870" ], [ "229", "5470" ], [ "381", "74" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:35", "updated_at": "2026-07-01 22:55:24" }, { "id": 143, "curve_key": "64149897:-513509658021", "ainvs": [ "1", "-1", "0", "-1336456", "594673996" ], "rank_lower_bound": 8, "naive_height": 53.93019913023835, "faltings_height": 2.247159462727608, "conductor": "311374862578586", "bad_primes": [ "2", "277", "3221", "174495029" ], "discriminant": "172501673868536644", "regulator": "5723.70755728757996648773821237143196032559827658967931", "points": [ [ "682", "-64" ], [ "-426", "-32750" ], [ "-149", "28190" ], [ "2451/4", "-21287/8" ], [ "2621", "-124160" ], [ "-980", "-30534" ], [ "5114", "354496" ], [ "1236", "28190" ] ], "submitter": "cocoxhuang", "commentary": null, "created_at": "2026-06-24 02:20:48", "updated_at": "2026-07-01 22:56:22" }, { "id": 142, "curve_key": "106566096:-1119953214144", "ainvs": [ "0", "0", "0", "-2220127", "1296242146" ], "rank_lower_bound": 8, "naive_height": 55.48861605455463, "faltings_height": 2.5144218111173267, "conductor": "255180258802520", "bad_primes": [ "2", "5", "6691", "953445893" ], "discriminant": "-25518025880252000000", "regulator": "2415.77505195822525458100130161611405965758066228127365", "points": [ [ "797", "5750" ], [ "597", "-13550" ], [ "1397", "30350" ], [ "1097", "13450" ], [ "907", "-5360" ], [ "-53", "37600" ], [ "897", "-5150" ], [ "1697", "49150" ] ], "submitter": "cocoxhuang", "commentary": null, "created_at": "2026-06-24 02:20:07", "updated_at": "2026-07-01 22:56:22" }, { "id": 29, "curve_key": "311738735319921:-5487609984705535174281", "ainvs": [ "1", "-1", "1", "-6494556985832", "6351402068900282539" ], "rank_lower_bound": 8, "naive_height": 100.1195596984162, "faltings_height": 6.158449360060919, "conductor": "72422714076260761748899710", "bad_primes": [ "2", "3", "5", "37", "151", "251", "5939", "96619896654233" ], "discriminant": "104878982358622652630973814596384000000", "regulator": "192112.135790124168751416663971415608442987571179367034", "points": [ [ "1634797", "320450201" ], [ "-2569043", "-2464593799" ], [ "1725397", "530370401" ], [ "1536847", "10928201" ], [ "-1037903", "-3459834799" ], [ "-947303", "-3413266399" ], [ "1536949", "13272713" ], [ "13057", "2503312241" ] ], "submitter": "David Renshaw", "commentary": "Found by Grunewald - Zimmert (1977). A historical rank ≥ 8 record, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:09", "updated_at": "2026-07-01 22:56:22" }, { "id": 57, "curve_key": "66576:-58791960", "ainvs": [ "0", "0", "1", "-1387", "68046" ], "rank_lower_bound": 7, "naive_height": 35.77903133765187, "faltings_height": 1.0365403415969128, "conductor": "1829517077483", "bad_primes": [ "3527", "518717629" ], "discriminant": "-1829517077483", "regulator": "1306.442211994994493978784223406033003476005702575198", "points": [ [ "-49", "135" ], [ "-47", "171" ], [ "-44", "209" ], [ "-43", "219" ], [ "-42", "228" ], [ "-34", "275" ], [ "-30", "287" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Elkies-Watkins, \"Elliptic curves of large rank and small conductor\" (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=7: [0,0,1,-1387,68046], |Delta|=1829517077483; their I-column records 71 integral x-coordinates. The submitted points certify rank >= 7 here.", "created_at": "2026-05-29 00:04:00", "updated_at": "2026-07-01 22:56:22" }, { "id": 124, "curve_key": "273936:-130823640", "ainvs": [ "0", "0", "1", "-5707", "151416" ], "rank_lower_bound": 7, "naive_height": 37.5619493443338, "faltings_height": 1.0973230208891636, "conductor": "1991659717477", "bad_primes": [ "154681", "12875917" ], "discriminant": "1991659717477", "regulator": "1172.532507747258989672291236129267860617335012322613", "points": [ [ "32", "39" ], [ "31", "65" ], [ "29", "101" ], [ "28", "116" ], [ "54", "26" ], [ "55", "62" ], [ "69", "293" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:30", "updated_at": "2026-07-01 22:56:22" }, { "id": 125, "curve_key": "287161:-142145965", "ainvs": [ "1", "0", "1", "-5983", "164022" ], "rank_lower_bound": 7, "naive_height": 37.70339493919055, "faltings_height": 1.1022310350387687, "conductor": "1005276094726", "bad_primes": [ "2", "269", "2347", "796141" ], "discriminant": "2010552189452", "regulator": "893.2081712006927339910880239250292484334567123159772", "points": [ [ "33", "36" ], [ "29", "108" ], [ "55", "18" ], [ "57", "66" ], [ "34", "-3" ], [ "7", "346" ], [ "56", "45" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:31", "updated_at": "2026-07-01 22:56:23" }, { "id": 44, "curve_key": "480576:-299721600", "ainvs": [ "0", "0", "0", "-10012", "346900" ], "rank_lower_bound": 7, "naive_height": 39.24822199037867, "faltings_height": 1.2446863772490515, "conductor": "382623908456", "bad_primes": [ "2", "47827988557" ], "discriminant": "12243965070592", "regulator": "4318.994211976314485322288857542918022772937787830133", "points": [ [ "-114", "82" ], [ "-111", "301" ], [ "-106", "466" ], [ "-102", "554" ], [ "-100", "590" ], [ "-98", "622" ], [ "-79", "803" ] ], "submitter": "David Renshaw", "commentary": "Lowest-conductor rank 7 curve in the Sage/Watkins elliptic-curves database: [0, 0, 0, -10012, 346900], conductor 382623908456. The submitted integral points were found by an exact search and certified independent by the site verifier.", "created_at": "2026-05-28 05:01:41", "updated_at": "2026-07-01 22:56:23" }, { "id": 126, "curve_key": "696217:-577740349", "ainvs": [ "1", "0", "1", "-14505", "667472" ], "rank_lower_bound": 7, "naive_height": 40.3602500169959, "faltings_height": 1.2029509488548138, "conductor": "1066284226102", "bad_primes": [ "2", "302213", "1764127" ], "discriminant": "2132568452204", "regulator": "933.7798366942692057650934863707616263994964142286268", "points": [ [ "65", "-14" ], [ "75", "16" ], [ "61", "72" ], [ "59", "104" ], [ "64", "16" ], [ "89", "244" ], [ "62", "55" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:32", "updated_at": "2026-07-01 22:57:20" }, { "id": 120, "curve_key": "707161:-600877405", "ainvs": [ "1", "0", "1", "-14733", "694232" ], "rank_lower_bound": 7, "naive_height": 40.4278029733335, "faltings_height": 1.2354028471005079, "conductor": "536670340706", "bad_primes": [ "2", "93463", "2871031" ], "discriminant": "-4293362725648", "regulator": "747.6205461568833835575943461684343776186191875739400", "points": [ [ "71", "50" ], [ "79", "118" ], [ "55", "198" ], [ "83", "170" ], [ "51", "250" ], [ "87", "226" ], [ "68", "55" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:25", "updated_at": "2026-07-01 22:57:20" }, { "id": 127, "curve_key": "747696:-643573080", "ainvs": [ "0", "0", "1", "-15577", "744876" ], "rank_lower_bound": 7, "naive_height": 40.57425527170349, "faltings_height": 1.2133517848099866, "conductor": "2206378706437", "bad_primes": [ "2206378706437" ], "discriminant": "2206378706437", "regulator": "1153.178388687324427112843454112369052995943965950190", "points": [ [ "68", "8" ], [ "76", "0" ], [ "67", "44" ], [ "77", "44" ], [ "66", "65" ], [ "78", "66" ], [ "63", "116" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:33", "updated_at": "2026-07-01 22:57:20" }, { "id": 123, "curve_key": "895881:-823856805", "ainvs": [ "1", "-1", "0", "-18664", "958204" ], "rank_lower_bound": 7, "naive_height": 41.11668861191012, "faltings_height": 1.338824330535397, "conductor": "896913586322", "bad_primes": [ "2", "13", "34496676397" ], "discriminant": "23319753244372", "regulator": "434.6225864840429444004928220928854851124416317346907", "points": [ [ "54", "298" ], [ "106", "350" ], [ "67", "38" ], [ "28", "662" ], [ "132", "818" ], [ "158", "1312" ], [ "15", "818" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:29", "updated_at": "2026-07-01 22:57:20" }, { "id": 121, "curve_key": "1760304:-2337014808", "ainvs": [ "0", "0", "1", "-36673", "2704878" ], "rank_lower_bound": 7, "naive_height": 43.14428045762337, "faltings_height": 1.3529721439201032, "conductor": "814434447535", "bad_primes": [ "5", "162886889507" ], "discriminant": "-4072172237675", "regulator": "584.4980643050451293253654768401361089199857192260729", "points": [ [ "111", "42" ], [ "106", "92" ], [ "121", "197" ], [ "86", "432" ], [ "136", "482" ], [ "109", "50" ], [ "71", "677" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:26", "updated_at": "2026-07-01 22:57:21" }, { "id": 122, "curve_key": "4447497:-9368130789", "ainvs": [ "1", "-1", "0", "-92656", "10865908" ], "rank_lower_bound": 7, "naive_height": 45.92355607195693, "faltings_height": 1.607028801729994, "conductor": "858426129202", "bad_primes": [ "2", "71", "607", "9959233" ], "discriminant": "121896510346684", "regulator": "507.9996880326439166864305051339475554028263290755461", "points": [ [ "168", "58" ], [ "239", "1407" ], [ "26", "2898" ], [ "97", "1620" ], [ "-258", "4318" ], [ "594", "12554" ], [ "-187", "4744" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:27", "updated_at": "2026-07-01 22:57:21" }, { "id": 128, "curve_key": "1066615641:-35239308179805", "ainvs": [ "1", "-1", "0", "-22221159", "40791791609" ], "rank_lower_bound": 7, "naive_height": 62.38636657025946, "faltings_height": 3.069903693575893, "conductor": "13077343449126", "bad_primes": [ "2", "3", "11", "23", "419", "6853501" ], "discriminant": "-16408200601321750438668", "regulator": "244.7081168123110517424651740118236827089076400521766", "points": [ [ "3055", "36043" ], [ "2296", "42115" ], [ "778", "154447" ], [ "1537", "100558" ], [ "272", "186325" ], [ "-3017", "284995" ], [ "1240", "122437" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:34", "updated_at": "2026-07-01 22:57:21" }, { "id": 26, "curve_key": "71280955234384:-568729306424497084352", "ainvs": [ "0", "-1", "0", "-1485019900716", "658252007072023716" ], "rank_lower_bound": 7, "naive_height": 95.69295090047126, "faltings_height": 5.918519997709521, "conductor": "1321186210228868621060280", "bad_primes": [ "2", "3", "5", "11", "13", "17", "19", "23", "29", "31", "37", "8420798017" ], "discriminant": "22409547184983924741173483181203078400", "regulator": "1031223.943702713970942638847459818408913490427960735", "points": [ [ "-1277354", "686274900" ], [ "2096456", "2599838590" ], [ "6341180", "15691521794" ], [ "5118316", "11275759890" ], [ "7761138", "21368830956" ], [ "6659304/25", "66330632742/125" ], [ "206059216/25", "2927181185214/125" ] ], "submitter": "David Renshaw", "commentary": "Found by Penney - Pomerance (1975). A curve of rank exactly 7, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:07", "updated_at": "2026-07-01 22:58:18" }, { "id": 27, "curve_key": "154832063750224:-1905911076227795039168", "ainvs": [ "0", "-1", "0", "-3225667994796", "2205916672708538820" ], "rank_lower_bound": 7, "naive_height": 98.02008655741527, "faltings_height": 6.036623018424112, "conductor": "2705139105695704399945080", "bad_primes": [ "2", "3", "5", "11", "13", "17", "19", "23", "29", "31", "37", "291701", "2186969" ], "discriminant": "45883723249375846665777287372882822400", "regulator": "486718.9016642236194594447475021625676457117043360219", "points": [ [ "1124958", "29035380" ], [ "707724", "526795542" ], [ "885716", "209112382" ], [ "333788", "1080007450" ], [ "1219101", "292124718" ], [ "6320106", "15305815332" ], [ "-1935671", "1094134272" ] ], "submitter": "David Renshaw", "commentary": "Found by Penney - Pomerance (1975). A curve of rank exactly 7, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:07", "updated_at": "2026-07-01 22:58:18" }, { "id": 28, "curve_key": "157907714207824:-1963828987407564843968", "ainvs": [ "0", "-1", "0", "-3289744045996", "2272951313488252420" ], "rank_lower_bound": 7, "naive_height": 98.07909567305349, "faltings_height": 6.039770338778831, "conductor": "2756084633910119906617080", "bad_primes": [ "2", "3", "5", "11", "13", "17", "19", "23", "29", "31", "37", "59", "11016191591" ], "discriminant": "46747845361418708281172757979630982400", "regulator": "403784.1579848618784679002566306921904067755443292495", "points": [ [ "1438008", "718248070" ], [ "870276", "262859618" ], [ "928733", "136859730" ], [ "1166620", "151132950" ], [ "15828084", "62574630898" ], [ "-99958904/49", "241216239846/343" ], [ "269482005/4", "4422204477635/8" ] ], "submitter": "David Renshaw", "commentary": "Found by Penney - Pomerance (1975). A curve of rank exactly 7, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:08", "updated_at": "2026-07-01 22:58:19" }, { "id": 56, "curve_key": "13296:-3945240", "ainvs": [ "0", "0", "1", "-277", "4566" ], "rank_lower_bound": 6, "naive_height": 30.376040693827097, "faltings_height": 0.5828325577145619, "conductor": "7647224363", "bad_primes": [ "7647224363" ], "discriminant": "-7647224363", "regulator": "159.18394480133901410572598227360268100133182426854", "points": [ [ "-22", "3" ], [ "-21", "33" ], [ "-19", "54" ], [ "-14", "75" ], [ "-12", "78" ], [ "-7", "78" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Elkies-Watkins, \"Elliptic curves of large rank and small conductor\" (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=6: [0,0,1,-277,4566], |Delta|=7647224363; their I-column records 49 integral x-coordinates. The submitted points certify rank >= 6 here.", "created_at": "2026-05-29 00:03:59", "updated_at": "2026-07-01 22:58:19" }, { "id": 117, "curve_key": "18192:-4468824", "ainvs": [ "0", "0", "1", "-379", "5172" ], "rank_lower_bound": 6, "naive_height": 30.625271689455815, "faltings_height": 0.5936988004113499, "conductor": "8072781371", "bad_primes": [ "8072781371" ], "discriminant": "-8072781371", "regulator": "150.95696885474745111489758710978373050933997082981", "points": [ [ "8", "51" ], [ "12", "48" ], [ "7", "53" ], [ "15", "53" ], [ "2", "66" ], [ "-1", "74" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:20", "updated_at": "2026-07-01 22:58:19" }, { "id": 119, "curve_key": "18736:-4830040", "ainvs": [ "0", "1", "1", "-390", "5460" ], "rank_lower_bound": 6, "naive_height": 30.780730614335926, "faltings_height": 0.6080932690520194, "conductor": "9694585723", "bad_primes": [ "9694585723" ], "discriminant": "-9694585723", "regulator": "198.72083733594502037312969609344942463062529685365", "points": [ [ "12", "51" ], [ "13", "52" ], [ "5", "60" ], [ "2", "68" ], [ "17", "63" ], [ "-4", "83" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:23", "updated_at": "2026-07-01 22:58:19" }, { "id": 118, "curve_key": "42672:-7905816", "ainvs": [ "0", "0", "1", "-889", "9150" ], "rank_lower_bound": 6, "naive_height": 31.983894739265388, "faltings_height": 0.6406966295363911, "conductor": "8796007189", "bad_primes": [ "8796007189" ], "discriminant": "8796007189", "regulator": "141.73365659632160880961097078808226407232604818979", "points": [ [ "12", "14" ], [ "11", "26" ], [ "10", "35" ], [ "22", "15" ], [ "23", "29" ], [ "25", "50" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:22", "updated_at": "2026-07-01 22:58:19" }, { "id": 113, "curve_key": "111657:-37043541", "ainvs": [ "1", "-1", "0", "-2326", "43456" ], "rank_lower_bound": 6, "naive_height": 34.86956085368328, "faltings_height": 0.7565833149092835, "conductor": "5739520802", "bad_primes": [ "2", "331", "383", "22637" ], "discriminant": "11479041604", "regulator": "98.157903581098532023878844268831034352354437276555", "points": [ [ "26", "-6" ], [ "30", "-14" ], [ "24", "20" ], [ "32", "16" ], [ "34", "36" ], [ "25", "9" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:15", "updated_at": "2026-07-01 22:59:16" }, { "id": 43, "curve_key": "123961:-43023725", "ainvs": [ "1", "1", "0", "-2582", "48720" ], "rank_lower_bound": 6, "naive_height": 35.18316683696271, "faltings_height": 0.8126178848463372, "conductor": "5187563742", "bad_primes": [ "2", "3", "2777", "311341" ], "discriminant": "31125382452", "regulator": "273.08550765251130993828811941017090645271641479597", "points": [ [ "-59", "35" ], [ "-58", "116" ], [ "-57", "150" ], [ "-53", "227" ], [ "-40", "320" ], [ "-32", "332" ] ], "submitter": "David Renshaw", "commentary": "Lowest-conductor rank 6 curve in the Sage/Watkins elliptic-curves database: [1, 1, 0, -2582, 48720], conductor 5187563742. The submitted integral points were found by an exact search and certified independent by the site verifier.", "created_at": "2026-05-28 05:01:18", "updated_at": "2026-07-01 22:59:17" }, { "id": 112, "curve_key": "339696:-203486040", "ainvs": [ "0", "0", "1", "-7077", "235516" ], "rank_lower_bound": 6, "naive_height": 38.262215921970544, "faltings_height": 1.0981908982946418, "conductor": "5258110041", "bad_primes": [ "3", "584234449" ], "discriminant": "-1277720739963", "regulator": "76.535132380495369224456093961543943630934899860102", "points": [ [ "46", "85" ], [ "55", "112" ], [ "64", "211" ], [ "43", "103" ], [ "19", "328" ], [ "82", "454" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:14", "updated_at": "2026-07-01 22:59:17" }, { "id": 116, "curve_key": "442897:-294730201", "ainvs": [ "1", "0", "0", "-9227", "340354" ], "rank_lower_bound": 6, "naive_height": 39.003277549050914, "faltings_height": 0.9339172453151712, "conductor": "6822208199", "bad_primes": [ "6822208199" ], "discriminant": "6822208199", "regulator": "207.82607512733307132827843581134934911794454010300", "points": [ [ "55", "-27" ], [ "54", "-10" ], [ "57", "-8" ], [ "49", "56" ], [ "62", "56" ], [ "89", "431" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:19", "updated_at": "2026-07-01 22:59:17" }, { "id": 114, "curve_key": "779961:-687300525", "ainvs": [ "1", "-1", "0", "-16249", "799549" ], "rank_lower_bound": 6, "naive_height": 40.7009975922472, "faltings_height": 1.1950656117869571, "conductor": "6601024978", "bad_primes": [ "2", "23", "1543", "93001" ], "discriminant": "1214588595952", "regulator": "65.565732131126563160853854716322922008539638659313", "points": [ [ "78", "7" ], [ "55", "237" ], [ "-14", "1019" ], [ "101", "375" ], [ "-106", "1203" ], [ "62", "135" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:17", "updated_at": "2026-07-01 22:59:17" }, { "id": 115, "curve_key": "3031041:-5241135105", "ainvs": [ "1", "-1", "1", "-63147", "6081915" ], "rank_lower_bound": 6, "naive_height": 44.773250048511805, "faltings_height": 1.5797551720520981, "conductor": "6663562874", "bad_primes": [ "2", "523", "6370519" ], "discriminant": "218351628255232", "regulator": "54.245797479930550808197728346032949250551289682168", "points": [ [ "171", "426" ], [ "107", "682" ], [ "155", "-70" ], [ "43", "1834" ], [ "203", "1162" ], [ "131", "146" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:18", "updated_at": "2026-07-01 22:59:18" }, { "id": 16, "curve_key": "1118815936:-37216742690816", "ainvs": [ "0", "-1", "0", "-23308665", "43082703225" ], "rank_lower_bound": 6, "naive_height": 62.50661028906994, "faltings_height": 3.0487468600126046, "conductor": "7727548608960", "bad_primes": [ "2", "3", "5", "17", "113", "769", "5449" ], "discriminant": "8906772526687296000000", "regulator": "419.30996796981054607112068235104029618565968205028", "points": [ [ "405", "183600" ], [ "-435", "230520" ], [ "2445", "26520" ], [ "4085", "126560" ], [ "695", "164980" ], [ "3705", "87000" ] ], "submitter": "David Renshaw", "commentary": "Found by Rignaux (1921). A curve of rank exactly 6, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:11:59", "updated_at": "2026-07-01 22:59:18" }, { "id": 20, "curve_key": "5903287504:-201055978351808", "ainvs": [ "0", "-1", "0", "-122985156", "232744673700" ], "rank_lower_bound": 6, "naive_height": 67.49632571023601, "faltings_height": 3.679756730921982, "conductor": "26799137200956120", "bad_primes": [ "2", "3", "5", "7", "11", "13", "17", "19", "23", "107", "467", "601" ], "discriminant": "95659142906961080877830400", "regulator": "1060.4130531684812274213611098178300928340852385953", "points": [ [ "1876", "92862" ], [ "-4892", "846930" ], [ "40616", "7889138" ], [ "10741", "388362" ], [ "13116", "935862" ], [ "311696/25", "99790306/125" ] ], "submitter": "David Renshaw", "commentary": "Found by Penney - Pomerance (1974). A curve of rank exactly 6, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:02", "updated_at": "2026-07-01 23:00:15" }, { "id": 17, "curve_key": "39951745024:-7933979669065568", "ainvs": [ "0", "-1", "0", "-832328021", "9183124281870" ], "rank_lower_bound": 6, "naive_height": 73.23281456521498, "faltings_height": 3.9492080152502926, "conductor": "1064348253719210460", "bad_primes": [ "2", "3", "5", "7", "11", "13", "17", "19", "23", "2385437629" ], "discriminant": "474897013203413671310840400", "regulator": "2701.6399282525630330343209741966428056184437506230", "points": [ [ "15541", "34307" ], [ "9333", "1492605" ], [ "7129", "1900453" ], [ "-186011/9", "113131711/27" ], [ "1637171/49", "1482112071/343" ], [ "1378809", "1618682523" ] ], "submitter": "David Renshaw", "commentary": "Found by Penney - Pomerance (1974). A curve of rank exactly 6, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:00", "updated_at": "2026-07-01 23:00:15" }, { "id": 21, "curve_key": "40906085584:-8217699530925248", "ainvs": [ "0", "-1", "0", "-852210116", "9511510378980" ], "rank_lower_bound": 6, "naive_height": 73.30363404213483, "faltings_height": 3.9568488879557058, "conductor": "5131476003533880", "bad_primes": [ "2", "3", "5", "7", "11", "13", "17", "19", "23", "29", "2179" ], "discriminant": "531185208959529592186118400", "regulator": "37742.286448328310471088273953984925467953157552812", "points": [ [ "-10234", "4142592" ], [ "15153", "277620" ], [ "-8324", "4003558" ], [ "54376", "11133058" ], [ "-128603/4", "15339115/8" ], [ "-177656/9", "116575766/27" ] ], "submitter": "David Renshaw", "commentary": "Found by Penney - Pomerance (1974). A curve of rank exactly 6, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:03", "updated_at": "2026-07-01 23:00:15" }, { "id": 22, "curve_key": "79900701904:-22546386352035008", "ainvs": [ "0", "-1", "0", "-1664597956", "26095909440100" ], "rank_lower_bound": 6, "naive_height": 75.31215142338309, "faltings_height": 4.06970936379526, "conductor": "284741255186508120", "bad_primes": [ "2", "3", "5", "7", "11", "13", "17", "19", "23", "319083769" ], "discriminant": "1016379901231367708305670400", "regulator": "1856.4874679486537029243421496659543977988291008021", "points": [ [ "24378", "57200" ], [ "-25000", "7217010" ], [ "37876", "4169262" ], [ "-12534", "6707512" ], [ "162250/9", "37661120/27" ], [ "2382832/9", "3636914050/27" ] ], "submitter": "David Renshaw", "commentary": "Found by Penney - Pomerance (1974). A curve of rank exactly 6, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:04", "updated_at": "2026-07-01 23:00:16" }, { "id": 23, "curve_key": "95071251664:-29278372602398912", "ainvs": [ "0", "-1", "0", "-1980651076", "33887665543876" ], "rank_lower_bound": 6, "naive_height": 75.83367739482048, "faltings_height": 4.099986335100618, "conductor": "337623153396448920", "bad_primes": [ "2", "3", "5", "7", "11", "13", "17", "19", "23", "378343729" ], "discriminant": "1205141092314624597939206400", "regulator": "1652.7003416966787588826095516412822645222470253356", "points": [ [ "24910", "76704" ], [ "24948", "38246" ], [ "20968", "1255254" ], [ "19074", "1745800" ], [ "10546", "3764640" ], [ "752437/4", "635863443/8" ] ], "submitter": "David Renshaw", "commentary": "Found by Penney - Pomerance (1974). A curve of rank exactly 6, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:04", "updated_at": "2026-07-01 23:00:16" }, { "id": 18, "curve_key": "99323902720:-31269101880111200", "ainvs": [ "0", "-1", "0", "-2069247973", "36191779888342" ], "rank_lower_bound": 6, "naive_height": 75.96495627364892, "faltings_height": 4.106342748418685, "conductor": "21417609338825700", "bad_primes": [ "2", "3", "5", "7", "11", "13", "17", "19", "23", "127", "10799" ], "discriminant": "1213641447728106672060786000", "regulator": "854.99378511946814432682065976442547356253336901804", "points": [ [ "28366", "564642" ], [ "41701", "4734687" ], [ "27337", "230775" ], [ "41709/4", "31747841/8" ], [ "85581", "22043153" ], [ "128029/4", "13266729/8" ] ], "submitter": "David Renshaw", "commentary": "Found by Penney - Pomerance (1974). A curve of rank exactly 6, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:01", "updated_at": "2026-07-01 23:00:16" }, { "id": 24, "curve_key": "114671748304:-38799289186774208", "ainvs": [ "0", "-1", "0", "-2388994756", "44907381038500" ], "rank_lower_bound": 6, "naive_height": 76.3960185638232, "faltings_height": 4.133029403084178, "conductor": "405947076034620120", "bad_primes": [ "2", "3", "5", "7", "11", "13", "17", "19", "23", "53", "997", "8609" ], "discriminant": "1449022372170745950888710400", "regulator": "2040.2613090018833134080325418276186635913655423722", "points": [ [ "27448", "112350" ], [ "29750", "405600" ], [ "-14000", "8695350" ], [ "-41724", "8482162" ], [ "117368608", "1271533747770" ], [ "-16300964/289", "2216801226/4913" ] ], "submitter": "David Renshaw", "commentary": "Found by Penney - Pomerance (1974). A curve of rank exactly 6, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:05", "updated_at": "2026-07-01 23:00:16" }, { "id": 25, "curve_key": "144101539024:-54673287071498432", "ainvs": [ "0", "-1", "0", "-3002115396", "63280268148996" ], "rank_lower_bound": 6, "naive_height": 77.08135206043404, "faltings_height": 4.1738204353785795, "conductor": "508534208459997720", "bad_primes": [ "2", "3", "5", "7", "11", "13", "17", "19", "23", "83", "409", "16787" ], "discriminant": "1815205696936306744772102400", "regulator": "2425.4713982456645308215970211688915508648574748522", "points": [ [ "32273", "76076" ], [ "24196", "2192250" ], [ "-45490", "10281544" ], [ "62528", "10955774" ], [ "127164", "41687310" ], [ "350114", "204765680" ] ], "submitter": "David Renshaw", "commentary": "Found by Penney - Pomerance (1974). A curve of rank exactly 6, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:06", "updated_at": "2026-07-01 23:01:13" }, { "id": 19, "curve_key": "157077339904:-62227614068304992", "ainvs": [ "0", "-1", "0", "-3272444581", "72023792282806" ], "rank_lower_bound": 6, "naive_height": 77.34001239556598, "faltings_height": 4.188573523568177, "conductor": "4330583892748773660", "bad_primes": [ "2", "3", "5", "7", "11", "13", "17", "19", "23", "29", "17614859" ], "discriminant": "1932244778818192209579608400", "regulator": "2971.4872960803653942500626665828891761232773805673", "points": [ [ "33609", "51175" ], [ "13009", "5626185" ], [ "22653", "3084991" ], [ "61026", "9979340" ], [ "57134", "8459010" ], [ "-216995/4", "75851127/8" ] ], "submitter": "David Renshaw", "commentary": "Found by Penney - Pomerance (1974). A curve of rank exactly 6, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:12:02", "updated_at": "2026-07-01 23:01:13" }, { "id": 55, "curve_key": "1057:-190801", "ainvs": [ "1", "0", "0", "-22", "219" ], "rank_lower_bound": 5, "naive_height": 24.31797255814547, "faltings_height": 0.08369007316794763, "conductor": "20384311", "bad_primes": [ "3089", "6599" ], "discriminant": "-20384311", "regulator": "23.0216507645579847918634719671775842289367147243", "points": [ [ "-7", "10" ], [ "-6", "15" ], [ "-5", "17" ], [ "-2", "17" ], [ "-1", "16" ] ], "submitter": "David Renshaw", "commentary": "Provenance: Elkies-Watkins, \"Elliptic curves of large rank and small conductor\" (arXiv:math/0403374), Table 4 low absolute-discriminant list for r=5: [1,0,0,-22,219], |Delta|=20384311; their I-column records 29 integral x-coordinates. The submitted points certify rank >= 5 here.", "created_at": "2026-05-29 00:03:58", "updated_at": "2026-07-01 23:01:14" }, { "id": 41, "curve_key": "3792:-295704", "ainvs": [ "0", "0", "1", "-79", "342" ], "rank_lower_bound": 5, "naive_height": 25.194228465945226, "faltings_height": 0.10809173841132348, "conductor": "19047851", "bad_primes": [ "19047851" ], "discriminant": "-19047851", "regulator": "14.7905275701311284815007921692282023510586664287", "points": [ [ "5", "8" ], [ "4", "9" ], [ "3", "11" ], [ "7", "11" ], [ "0", "18" ] ], "submitter": "David Renshaw", "commentary": "Found by Brumer - McGuinness. A rank 5 example from their elliptic-curve examples page, with independent points of x-coordinates 5, 4, 3, 7, 0.\r\n\r\nThis curve has rank exactly 5 (verified by both mwrank and the gp function ellrank().", "created_at": "2026-05-28 04:35:25", "updated_at": "2026-07-01 23:01:14" }, { "id": 110, "curve_key": "4816:-124120", "ainvs": [ "0", "1", "1", "-100", "110" ], "rank_lower_bound": 5, "naive_height": 25.43909696096597, "faltings_height": 0.17920167927724356, "conductor": "55726757", "bad_primes": [ "647", "86131" ], "discriminant": "55726757", "regulator": "23.8551122555658031101980388427178666066140386433", "points": [ [ "1", "3" ], [ "0", "10" ], [ "-1", "14" ], [ "-2", "17" ], [ "-4", "21" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:11", "updated_at": "2026-07-01 23:01:14" }, { "id": 111, "curve_key": "6672:-632664", "ainvs": [ "0", "0", "1", "-139", "732" ], "rank_lower_bound": 5, "naive_height": 26.715389509023797, "faltings_height": 0.21477646545169604, "conductor": "59754491", "bad_primes": [ "409", "146099" ], "discriminant": "-59754491", "regulator": "21.6690124194189046816235098629475654280180199560", "points": [ [ "8", "11" ], [ "4", "15" ], [ "9", "14" ], [ "3", "18" ], [ "10", "18" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:13", "updated_at": "2026-07-01 23:01:14" }, { "id": 149, "curve_key": "8112:-803736", "ainvs": [ "0", "0", "1", "-169", "930" ], "rank_lower_bound": 5, "naive_height": 27.194052272061576, "faltings_height": 0.23405792705071166, "conductor": "64921931", "bad_primes": [ "64921931" ], "discriminant": "-64921931", "regulator": "20.3987884109559936469620892946805796036044812187", "points": [ [ "8", "9" ], [ "7", "9" ], [ "6", "11" ], [ "0", "30" ], [ "20", "74" ] ], "submitter": "Seewoo Lee", "commentary": "Test submission, https://www.lmfdb.org/EllipticCurve/Q/64921931/a/1", "created_at": "2026-06-24 18:11:42", "updated_at": "2026-07-01 23:01:15" }, { "id": 107, "curve_key": "11856:-1275480", "ainvs": [ "0", "0", "1", "-247", "1476" ], "rank_lower_bound": 5, "naive_height": 28.141768042607605, "faltings_height": 0.2182276673287099, "conductor": "22966597", "bad_primes": [ "47", "488651" ], "discriminant": "22966597", "regulator": "13.8258999407123215574851393149106059573573060505", "points": [ [ "8", "3" ], [ "10", "2" ], [ "7", "9" ], [ "11", "9" ], [ "6", "14" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:07", "updated_at": "2026-07-01 23:02:11" }, { "id": 108, "curve_key": "19929:-2917917", "ainvs": [ "1", "-1", "0", "-415", "3481" ], "rank_lower_bound": 5, "naive_height": 29.772761127041008, "faltings_height": 0.40394763752598917, "conductor": "34672310", "bad_primes": [ "2", "5", "79", "43889" ], "discriminant": "-346723100", "regulator": "7.98812089828060622749800735401369801305082865558", "points": [ [ "8", "21" ], [ "13", "6" ], [ "18", "31" ], [ "3", "46" ], [ "12", "5" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:09", "updated_at": "2026-07-01 23:03:08" }, { "id": 109, "curve_key": "25536:-3818880", "ainvs": [ "0", "0", "0", "-532", "4420" ], "rank_lower_bound": 5, "naive_height": 30.443533500748607, "faltings_height": 0.48866398864488353, "conductor": "37396136", "bad_primes": [ "2", "4674517" ], "discriminant": "1196676352", "regulator": "10.3187357247916407891471609820307434164395313639", "points": [ [ "10", "10" ], [ "18", "26" ], [ "16", "2" ], [ "6", "38" ], [ "8", "26" ] ], "submitter": "David Renshaw", "commentary": "From the tables of Elkies-Watkins 2004, 'Elliptic curves of large rank and small conductor' (ANTS-VI, arXiv:math/0403374): a low-conductor / low-discriminant curve of this rank that was absent from the leaderboard. Witness: a maximal independent set of rational points (Neron-Tate height pairing).", "created_at": "2026-06-24 00:08:10", "updated_at": "2026-07-01 23:03:09" }, { "id": 72, "curve_key": "336:-31320", "ainvs": [ "0", "0", "1", "-7", "36" ], "rank_lower_bound": 4, "naive_height": 20.70402430020948, "faltings_height": -0.2176951115207691, "conductor": "545723", "bad_primes": [ "545723" ], "discriminant": "-545723", "regulator": "3.306378474591305516092927336098302768950667315", "points": [ [ "-4", "0" ], [ "-3", "5" ], [ "-2", "6" ], [ "-1", "6" ] ], "submitter": "David Renshaw", "commentary": "Provenance: found by a local bounded-height search over reduced integral Weierstrass models. Invariants are c4=336, c6=-31320, Delta=-545723 with 545723 prime. The submitted points (-4,0), (-3,5), (-2,6), and (-1,6) were checked exactly on the curve, with an additional finite-field independence sanity check; the site Neron-Tate verifier gives the definitive certificate.", "created_at": "2026-06-01 13:36:55", "updated_at": "2026-07-01 23:03:09" }, { "id": 71, "curve_key": "3616:137440", "ainvs": [ "0", "-1", "0", "-75", "-134" ], "rank_lower_bound": 4, "naive_height": 24.5793711645362, "faltings_height": 0.08614869463915334, "conductor": "16430032", "bad_primes": [ "2", "353", "2909" ], "discriminant": "16430032", "regulator": "29.93011794279284801770139339930493926616246710", "points": [ [ "-2", "2" ], [ "-6", "8" ], [ "10", "4" ], [ "30", "154" ] ], "submitter": "David Renshaw", "commentary": "Provenance: found by a local bounded-height search over reduced integral Weierstrass models. Invariants are c4=3616, c6=137440, Delta=16430032 = 2^4*353*2909. The submitted points (-2,2), (-6,8), (10,4), (30,154) were checked exactly on the curve, with an additional finite-field independence sanity check; the site Neron-Tate verifier gives the definitive certificate.", "created_at": "2026-05-29 13:46:05", "updated_at": "2026-07-01 23:03:09" }, { "id": 54, "curve_key": "3801:-232605", "ainvs": [ "1", "-1", "0", "-79", "289" ], "rank_lower_bound": 4, "naive_height": 24.729058406967745, "faltings_height": -0.08697563887120269, "conductor": "234446", "bad_primes": [ "2", "117223" ], "discriminant": "468892", "regulator": "1.504344888275283974095271252282834672518585514", "points": [ [ "6", "-1" ], [ "4", "3" ], [ "5", "-2" ], [ "8", "7" ] ], "submitter": "David Renshaw", "commentary": "Smallest conductor of an elliptic curve of rank 4: conductor 234446 = 2·117223, Cremona label 234446a1 / LMFDB 234446.a1, y^2 + x·y = x^3 − x^2 − 79x + 289. By completeness of Cremona's tables below conductor 500000, this is the provably minimal conductor in rank 4. Generators [6,−1], [4,3], [5,−2], [8,7].", "created_at": "2026-05-28 21:07:17", "updated_at": "2026-07-01 23:03:09" }, { "id": 14, "curve_key": "1183168:-1163801600", "ainvs": [ "0", "-1", "0", "-24649", "1355209" ], "rank_lower_bound": 4, "naive_height": 41.951118434136355, "faltings_height": 1.4675522067412143, "conductor": "13217088", "bad_primes": [ "2", "3", "23", "41", "73" ], "discriminant": "174691415199744", "regulator": "24.11485382784266345603218697563936280642992865", "points": [ [ "40", "657" ], [ "21", "920" ], [ "-55/16", "76797/64" ], [ "3305/121", "1114848/1331" ] ], "submitter": "David Renshaw", "commentary": "Found by Wiman (1945). A curve of rank exactly 4, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:11:58", "updated_at": "2026-07-01 23:03:10" }, { "id": 15, "curve_key": "41134419648:0", "ainvs": [ "0", "0", "0", "-856967076", "0" ], "rank_lower_bound": 4, "naive_height": 73.32033320698703, "faltings_height": 4.178268178890409, "conductor": "13711473216", "bad_primes": [ "2", "3", "7", "17", "41" ], "discriminant": "40278416178577537588263358464", "regulator": "172.7920313416794830582707377111743516153900428", "points": [ [ "-4998", "2039184" ], [ "34476", "3381300" ], [ "53958", "10528848" ], [ "1843559424/25", "79156334875968/125" ] ], "submitter": "David Renshaw", "commentary": "Found by Wiman (1945). A curve of rank exactly 4, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:11:58", "updated_at": "2026-07-01 23:03:10" }, { "id": 70, "curve_key": "273:999", "ainvs": [ "1", "-1", "1", "-6", "0" ], "rank_lower_bound": 3, "naive_height": 16.82841538555488, "faltings_height": -0.5324408450204543, "conductor": "11197", "bad_primes": [ "11197" ], "discriminant": "11197", "regulator": "0.86842195369372664900484860186367158964425101", "points": [ [ "0", "0" ], [ "-1", "2" ], [ "-2", "1" ] ], "submitter": "David Renshaw", "commentary": "Provenance: found by a local bounded-height search over reduced integral Weierstrass models. Invariants are c4=273, c6=999, Delta=11197. The submitted points (0,0), (-1,2), (-2,1) were checked exactly on the curve, with an additional finite-field independence sanity check; the site Neron-Tate verifier gives the definitive certificate.", "created_at": "2026-05-29 13:46:04", "updated_at": "2026-07-01 23:04:07" }, { "id": 53, "curve_key": "336:-5400", "ainvs": [ "0", "0", "1", "-7", "6" ], "rank_lower_bound": 3, "naive_height": 17.451333479889612, "faltings_height": -0.5613901422939866, "conductor": "5077", "bad_primes": [ "5077" ], "discriminant": "5077", "regulator": "0.41714355875838396981711954461809339674981011", "points": [ [ "1", "0" ], [ "2", "0" ], [ "0", "2" ] ], "submitter": "David Renshaw", "commentary": "Smallest conductor of an elliptic curve of rank 3: conductor 5077 (prime), Cremona label 5077a1 / LMFDB 5077.a1, y^2 + y = x^3 − 7x + 6. Cremona's tables are complete below conductor 500000, so 5077 is the provably minimal conductor in rank 3. Generators [1,0], [2,0], [0,2].", "created_at": "2026-05-28 21:07:16", "updated_at": "2026-07-01 23:04:07" }, { "id": 13, "curve_key": "3936:0", "ainvs": [ "0", "0", "0", "-82", "0" ], "rank_lower_bound": 3, "naive_height": 24.83376077451643, "faltings_height": 0.1377204761845264, "conductor": "430336", "bad_primes": [ "2", "41" ], "discriminant": "35287552", "regulator": "10.207892029767887379642364292744681701782086", "points": [ [ "-8", "12" ], [ "-1", "9" ], [ "49/4", "231/8" ] ], "submitter": "David Renshaw", "commentary": "Found by Billing (1938). A curve of rank exactly 3, via Dujella's elliptic-curve rank-records tables.", "created_at": "2026-05-27 22:11:57", "updated_at": "2026-07-01 23:04:07" }, { "id": 69, "curve_key": "1:-865", "ainvs": [ "1", "0", "0", "0", "1" ], "rank_lower_bound": 2, "naive_height": 13.525459013863758, "faltings_height": -0.8152203271973504, "conductor": "433", "bad_primes": [ "433" ], "discriminant": "-433", "regulator": "0.224694163418166741612071133995500441366829", "points": [ [ "0", "1" ], [ "-1", "1" ] ], "submitter": "David Renshaw", "commentary": "Provenance: found by a local bounded-height search over reduced integral Weierstrass models. Invariants are c4=1, c6=-865, Delta=-433. The submitted points (0,1), (-1,1) were checked exactly on the curve, with an additional finite-field independence sanity check; the site Neron-Tate verifier gives the definitive certificate.", "created_at": "2026-05-29 13:46:04", "updated_at": "2026-07-01 23:04:07" }, { "id": 2, "curve_key": "112:-856", "ainvs": [ "0", "1", "1", "-2", "0" ], "rank_lower_bound": 2, "naive_height": 14.155496613885283, "faltings_height": -0.7956416542942529, "conductor": "389", "bad_primes": [ "389" ], "discriminant": "389", "regulator": "0.152460177943143751624324757049455823243727", "points": [ [ "0", "0" ], [ "-1", "1" ] ], "submitter": "David Renshaw", "commentary": null, "created_at": "2026-05-27 22:07:10", "updated_at": "2026-07-01 23:04:07" }, { "id": 42, "curve_key": "48:-216", "ainvs": [ "0", "0", "1", "-1", "0" ], "rank_lower_bound": 1, "naive_height": 11.613603032723672, "faltings_height": -0.9965422076373671, "conductor": "37", "bad_primes": [ "37" ], "discriminant": "37", "regulator": "0.05111140823996884023588609975694202160954", "points": [ [ "0", "0" ] ], "submitter": "David Renshaw", "commentary": "The standard conductor 37 rank-one curve y^2 + y = x^3 - x, with generator [0, 0].", "created_at": "2026-05-28 04:35:40", "updated_at": "2026-07-01 23:04:08" } ] }