curve #1393

y² + xy = x³ − 11532891443929352092867911758070963219260449885x + 477129965035818985427866235650134910745513387031938136016451076770225

a-invariants

[1, 0, 0, -11532891443929352092867911758070963219260449885, 477129965035818985427866235650134910745513387031938136016451076770225]

rank (lower bound)

≥ 28

torsion subgroup

trivial

conductor (N)

4997539447084604265015693653254416670885537251925261031237083539217290415193756279081256913681258637880707649270

discriminant (Δ)

-172526987195500245437889701098090108871004464092403575685298243446871769421269159568154220077452472431447939224976859387240205332480000000

Faltings height

25.2520

naive height

329.8000

primes of bad reduction

2, 3, 5, 7, 13, 17, 23, 37, 101, 380869, 27746524433989861, 7160854931634141712827468527700751, 16555589013770658824199467146066474032839750983

regulator

5021017574554074272957688056276.070612276864145570149832558688836761170832047605286306248081930

submitted by

Ritik Jain

submitted at

2026-09-24 21:25:39 UTC

later contributions

primes of bad reduction recorded · wgxli · 2026-09-24 22:29:50 UTC

Witness: 28 independent points log in to add more points →

(9979786642567883558277080/169, 3116475935977456868395509029770118405/2197)
(582508810959439436416053575/7396, 4817191424263864681080482873823234929335/636056)
(168408237839786140518710834/25, 2185194713762723409699713228648175190963/125)
(299132560911255842434370, -154252178628365940192095606050295185)
(8265340719925969487157650/121, -3780287283669568725206379494149968035/1331)
(62812359289719005714870, -735699649529220287672796992444185)
(-49413902549820718668542830/841, 752476838000281127281235704064777825035/24389)
(70488428737934341052066, 3797984395572601399621306169456687)
(63827392553176919046710, 1021905299565977464790242645909475)
(546526078512949807343048330/12769, -11353946833290380749654589357211254009945/1442897)
(106987519251235939919270, 21630277237609172934407113791440915)
(153205259722909180762370, -48023342840869934879750655232019185)
(768809376614381610036557654/12769, -1450913139615677490729753182735706389801/1442897)
(441160050354156421526570, -285041776145234400524920411301769385)
(247563866682346925674446026/361, 3850068255497138092072476947102612629013/6859)
(-11116347102077067365653401100904158/247716239521, 3707563435298263889893302736104032219751288757972417/123291097288236431)
(1708268535666554864384949391350954920434821090395178770/29225646417428739622603764858721, -260568495998290545048253704640524286614877402168407956309656968441567527181232015/157996035136488290927885019145876658461943665519)
(-115728805870556383791710, 16181585538631593904061205796821295)
(-1028316460311646277162583471842990586660333130426/17452974274699059962119489, 2249810849314130831229618062481034284074357316483377473591366309116732499/72912866562598474026932322579554016737)
(5931905638466656868745042554191856480/94871671363681, 633342198595038322194540977113651160906637974067810935/924069907261567949329)
(8392645180003754718436109837841290931730815602879570/134146628548899965312354959801, 33946837109537453211359858312959979150939636627396629763303615880603631709635/49132588297351593686881183822837554183701549)
(177768284713382319354116397775634115735890/2675414513678574721, 8944574185156360608124446339250069221140977151303934565593905/4376093756681092140217651519)
(-1987176157124083401063591186177571711870/17943611693490289, 47837218615437240537468417500658829214974637228851491904345/2403614379928273589468663)
(-16922538203612543970324430/361, 207448921641428140075251341036342110085/6859)
(-17538601447147195999192631668965480546490268006530/189636023521352390840016721, 71644749667541744884252420589219713773200984297045732519514962070133125985/2611447263894709351831072352507852708519)
(307744484698286157201178785049469360/34123718085601, 3854212330153243679397865647633399398121029523323297065/199335439506660607151)
(13429172126449206276472636219520543715660309970/222436366598680969059081, 99973960323999665112626639184518934329139831733435895231338812638635/104908019590342573925613479810672229)
(46733138254919517787476338974580090/1778283591529, 32875633510554447638488812444089971506636475119318325/2371382069826526667)

Commentary

Specialization at T = 5036/209 of the rank-17 fibration of X1092 given by: y^2 - (Lx+B)y = x(x+D)(x+E), where B = pq(L+p+q) - pE - qD, L = 1299T^2 - 3766960T + 321877, p = 1830T^2 - 1401T - 19560, q = 24906T^2 + 1589093T - 8337493, D = 7(11896740T^4 - 356491276T^3 - 2949777123T^2 + 37369005238T - 96823708920), E = 10(57046374T^4 - 254317076T^3 - 379408012933T^2 - 35060842895T + 1673269025580).

last edited by wgxli at 2026-09-24 22:52:21 UTC · history