curve #728

y² + xy = x³ − 2695272471630565276270102221032380764896680795950115x + 58070645430746823930939039519345796725516724439864925595007798344169481404417

a-invariants: [1, 0, 0, -2695272471630565276270102221032380764896680795950115, 58070645430746823930939039519345796725516724439864925595007798344169481404417]
rank (lower bound): ≥ 29
torsion subgroup: trivial
conductor (N): 373311163292677499812856331134685180923731435354851746325416047107220696523044284065917244788194219682890915991886062890
discriminant (Δ): -203683787517950029015641023933573057569787662504242280824650680870198148366064409224537444616773975121893041403858152661840353838880789692513043986278400000
Faltings height: 28.5430
naive height: 367.0342
primes of bad reduction: 2, 3, 5, 7, 13, 19, 23, 37, 41, 47, 53, 61, 71, 83, 230356944105531027122959977046222329093333893244114716488245495013109908196343024498561458924446725619
regulator: 105924260477161814835722637714334.949314847667879625172769483442393999499957052771794895459192747
submitted by: wgxli
submitted at: 2026-09-14 00:43:54 UTC
last updated: 2026-09-14 00:43:54 UTC

Witness: 29 independent points log in to add more points →

(418228045847034807466938119/100, -216498069910260140225513788388659900580167/1000)
(296583835070036242594215067407013376276/1346567613889, -4977994223129867221543962105171066278555666635678559682941/1562579950806231713)
(-7456595017152713971258998838/361, -2221829304696731408903281829987016400116553/6859)
(445839830719056887104704050463406356574751169858157765195610330588465940254/11263257399577928892691333873534224611666087139241, 4376360753220120740920948316647695290546000574987858748672812820986970660477583104118821365621738999805106894563/37800366771852464300077018301857452323992163396256373292729439795761215211)
(145498066503447182598788658062/3721, 25342206394972379599412294660978851733419805/226981)
(-77053846024127683664568034/25, 32197780004063811684894325729457304916549/125)
(195827869811800141122678815145831017301023100334/564148886284765950921, -85744734567535592425784974724874054135355926440555506942853767696605283/13399562004222151060599669470469)
(148227069854399566677736939872871761038/4588965142969, 674795440335802360674712300120803601527504671936054551739/9830421472721333203)
(22294131232137657868159104242/841, -1767187328048226187641898101614956063401995/24389)
(412197937120441937269620808526/49, 264636943195775605846027024015885387502949359/343)
(120714844111282213322688243748471960217742074/30963174095946481, -1326180752204473605797430814515050204331603235376586842870984472713/5448390311859350270285879)
(630823876550428613256184106/25, 9784258532578550643656524163413272080689/125)
(-1436578199039549869592898001843185778/23907035161, 204298529249183561905385003911441910362408483269901867/3696481869558659)
(235602940633417491671513942, 3535686669914432079910731241224869289425)
(919022722796371359277760566835850464740586/5031421967955289, -849058000476237061595334676536224887828249606449949137272208931/356891424276161992232413)
(-206167132815886509744701983050653357374/5728713001729, -4515615756346322657777560864313777788267365005642702081359/13711519894387314817)
(-6403214093940613819431024195593254016/251471157961, 41860603869166272003825986469090745610734644019631050747/126104990111544709)
(-419666408329073174770069778368574494/21341411569, 1002852063632863611846551402044164092311185360316550539/3117702791880503)
(9208334135100011428405039127060719166/255513063289, 11365157557068600541440215697352603371488362538220627371/129157509770513587)
(10346670051343068194847404251444845159368029838/554730251081025776761, 1561732161579154401957318800553559317233452926282973294283812462508981/13065401941805264230293791425741)
(114036816367547089005563694647919168151614379/3165451276229052196, 495069706908325240012771870188282491481811063253821717720501886047/5631880739621675006342344744)
(150957311350027199526385164291294535622/938482375009, 1769090086961785421961089041747062803527272246601373550097/909157616237093777)
(28790391557987780963450314638312468536954/140048739339961, 4743575070381672842110359123057132694385677278818991400607347/1657367452023013925459)
(6463835327871789700300635836800388629967/720437673796, -519670629830195818574192844491013012742723295077794799053347/611497411390611656)
(5963891951007101872456053200996694480544346/165489570081552649, -5923618503905141394697582533792196642705795756101383034346008379/67321834457985961408192357)
(10622300491019295346894196077209892659217223504/32979904816883795761, 1081430462927195464015588322128080645294815313020459119559678498738777/189397436641752518371546540759)
(587013129845936723406815874975873042749/2239584082576, 13963960503363394832809215202310375542169300676753727722437/3351591329592965824)
(54108091759208421579831463712316274742332604146961760836/1431658306561277346535895776489, 172919964109868089540969558096251369812355429121563820295047542140283044092703236669/1713006717803165136761304003496736382423793013)
(166353119525545432673001793653695329358782/4277221345419961, -30738930356112292854354814405301962582809670016231634144927563/279732282007276451361709)

Commentary

Specialization at T = 20920/2293 of the rank-17 fibration of X1092 given by: y^2 + (Lx+B)y = x(x+D)(x+E), where B = pE + qD + pq(L-p-q), L = 33259T^2 + 122264T - 682370, p = 95175T^2 + 2786613T - 32668, q = 1936T^2 - 62631T + 606464, D = 7(133750152T^4 + 585493575T^3 - 46370601212T^2 + 50749834445T - 26401212000), E = 217588848T^4 - 1500919481T^3 - 129911904187T^2 + 1875481523886T + 337787445024.

last edited by wgxli at 2026-09-17 17:25:21 UTC · history