curve #1998

y² + xy = x³ − 545046419988593552431895214438043430543353952260x − 79854040664558700615260834076268034687822354920280972737190967916351984

a-invariants: [1, 0, 0, -545046419988593552431895214438043430543353952260, -79854040664558700615260834076268034687822354920280972737190967916351984]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 34910793879807212391975584054857751022998385247520531253870511025704633110375003514482
discriminant (Δ): 7608159007614361312334814838607448874532396657641327781439424382982184627373721524245604800286971997654866061827552810663466560979239007845097472
Faltings height: 26.4968
naive height: 341.3652
primes of bad reduction: 2, 3, 11, 17, 23, 37, 67, 71, 73, 83, 109, 193, 367, 499, 509, 709, 883, 1987, 4937, 5051, 7333, 52057, 148501, 256471, 374639, 502259, 571721, 579707
regulator: 27077046197927163246489048491369518.286518874193327304210771423342747011052
submitted by: Steps Unbounded
submitted at: 2026-09-26 04:55:19 UTC
last updated: 2026-09-26 04:55:19 UTC

Witness: 18 independent points log in to add more points →

(-2020406596452193686561905700556237650353285033212/8089068812336781561998161, 4641373441202616709010580227124380636753037290551036131322714646351970764/23006353832236438545896888225200170841)
(-723926582101116727961663161161821122197352642520409174136/2973306714292516171155463704949369, 31778032211790215011478691983491312418577375914446116267343869106655129236373593591564/162128575672543716137390934264907391922866246664653)
(-1044659937208442250266965335634564049296117884495434459726/1783707449703318968408348465984049, 14776434715101394674434578256238742972471550638274230552230540711958957225955380305448/75333030611800022770593566898135233254261577863593)
(-5360944097742807783449076276733889520311541950009930062279517482456/9248170457301159391116965808505396305030625, 5716321065174910453849846576209910740228832091326916382328404678506005234769992217563740112726840028/28124430610021289253650439944293674222686907904598719844932140625)
(351433568162719465371068799326671031592219012499529918871702042855662438864/310754696525566402474120438089076778325858856855249, -4744490157763052107966239736985048161259603641561528140474143152168501958728835255185848139473054852458475285668/5478057058422942003777575176828598877845084533517996763401519386279581500057)
(863745104418439539788569304317218967323186295950088/229555271433528046738405881, -24872229705346023827336953040672472279520098081192498025009533084566945550604/3478010632409630304102526490457145057779)
(10542904347374965673816173781012879137733331223593724282458210494/1743829743710368406709238990343640025, -1082531135377784719791322786803740200002452095853395807145258292853153872026729911252493438296018/2302799473196416673498369464360422526330030010702699875)
(120236099596487484808690065123058189472922564240412007039425890369861482456/65007697285934048373121776751933995954469726745161, -1199722494974028543569528428476259797130047234692211950102300636729385575267507086019267121391333280325231380692/524139842649908851494499483212024098267398905181523107611498336483807206341)
(2771885262883878398743199798998497229801427802649521196423535574433176655897484/2493951325093780180079333351450743043135099316787009225, -3265236936780408967337737943401167850872611791506369738209039660539636447862121258247446961544315063076948319620911972/3938510063808347722054332809463309843910738335979351877464367793592423569001269125)
(-2315621217676296285839556512940308181835329241510805688538333911309604/14609835726083355811124413881659821781827778641, 89221633847040534417959320829641680789524544661189080221999517917768105337069905349180487283971373161484/1765907695307243301472259833041284111062651070884104217673907687593689)
(-681218498286024474654073811000472663963181536170900582396/1141413827826182176904793725409849, 6990006737568072983727716984892119851498446159480557502279988048892199680243519567216/38562456755554747043047119984359476787868767051757)
(-2341545251658583076658371295968229563739665065672709932793225396362842165784/3722879264608758809285405963041460755502107684101241, -27017938569929852874576906354817955858953272822291331553981772367614921490596436196214070303548907157676474432404/227152974420748079328192378434843816640969612655613173088817933433977318624461)
(685704447602568102737274596469004618002278622210309092484369426259833690580440538861408958/676795600914113268536851072251749493268435322046688980758226001561, -355617207859519370542910117736879334260661620266018552694424548646475957023901510926101494179788273026197282387024559330730008898044872/556783415484580927427296615182432086223757200507321286168546630409977321652613275183519523718298941)
(138807305388103379256993448568561792430036152258692144961453969491450071425150484301670540500756634894/139176701751295011069474270718313701539039786640552356250977584015645445097249, -31523201730285181262054104483243967583808773322906383271337130259138046135831102522880477021440060241534891208471838624706409452398884722521121648735208/51921808417376769992316386634160295918861293457030205456569339958317806296666275070400017055776771534953283302059057)
(578141030890195427350513754027186164006663937918057988882790357531462317382803176456864/646407352408182840808266619780633420377837076782112744930646689, 6325010503212900333774550800686963462253182762470768373922902910571025199297241078905066391682860919343927104086751213595391691644/16434611090143021170459109113469187354253904862769014230164320333258071585147511969881121568113)
(293340890127911435968881771883701348896583849889703818527787097780353876/80725302945751674857830211525468630957943931521, -155427630676773967331980407405637207545624801641743127620340750770962863477018875546584299013702964572243628/22935833395837445217671843948154255864665415403991975823842571882464319)
(10128220045518608321208047514782563215171722460132454682193496021361517307688/786182508527676922161225960103073310772151221339081, -1017600361489321721145808349070135755581454795555271292316637365966314659484588707824423197534419269587789896157612/22043729123373170893695091413744893004937978021924065406515450563393973547771)
(33972948115517909670791634489364577826089306209133310711814252688649988821201981096422288/5756025238287503879054374028926012730417557607285851031391147601, -6211403496008482437839955833070731457162678668008586919827399704779727712297669338073246377721362908200759627816748245137465089803356/436700847007069901970130951181071305048310287971170555240225684911094010666487983524403525850151)

Commentary

2-isogenous partner of curve#506 (Matthias Breddin).

last edited by Steps Unbounded at 2026-09-26 21:35:56 UTC · history