curve #416

y² + xy = x³ − 280245422569286389673485906779759546676443623212141473025x + 1766436728610859597051807326576217018574447760859287591309968524822782669465597775625

a-invariants: [1, 0, 0, -280245422569286389673485906779759546676443623212141473025, 1766436728610859597051807326576217018574447760859287591309968524822782669465597775625]
rank (lower bound): ≥ 27
torsion subgroup: trivial
conductor (N): 2010216859852031560349698848399916716713771291115989029549967841193383632802929212562369436697784583982578261631109192638902465162773631018570
discriminant (Δ): 60656490490005214501764462674862504178121595585158962155203002008465561213160134917504974026228944875883385236630694301058504432649839906420906710401195422138178816000000
Faltings height: 31.3616
naive height: 401.5394
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 31, 61, 709, 50291, 655901, 126586399, 90726663499557641668127723, 6936584227069036800070792280872049982039628294132261241300860293698613565568464489397
regulator: 4886469518281632877642435838703002.7866452903600788780654245520177880219570197934484908704321
submitted by: 7fff-zip
submitted at: 2026-08-29 12:32:49 UTC
last updated: 2026-08-29 12:32:49 UTC

Witness: 27 independent points log in to add more points →

(-19133890401504111907105478590, -351559960864154060624187827459813170328485)
(-6777000465751928526676449550, -1831504246318298219216901386792540688835525)
(254571456454283392558298257732, 128173090905928666004765666608944803003167777)
(-13444255964235942952280273698, -1761847126629030558591774162555084481704673)
(59860409169533463232133955100/9, 11975150946124491153984461505751582430423875/27)
(-1168661919647208813850179700, 1446496718950822966619293413429303001769225)
(11853242401111975088931156050, -331651503965528261377633572429261491368525)
(-19283280338596462743304277890, 9915369362579932017761578235681291669315)
(5495913591443630786245994331290, -12884211091339945894100163510398009957616135925)
(116067393708208285972456320950, 39151720409666871187445888930897826471439475)
(70863538065380280800256039050, -18378207329236189309534989807823060824289525)
(25174219240799981315582710730, 3265795432949385666607521455930858614896875)
(267919906328665549300653684830, -138413374797951534876879082557848925110672125)
(30958493188088833645255955375/4, -1976823348964913323615219378487891099688575/8)
(-10644423314918775998556547150, 1882401281540311278976947699367799334124475)
(8253453748147722987380866100, -125160807657847997579663480489706155481775)
(13112472527031855549873780650, -588425586662598269460430532373237214055125)
(46219802422821914458384975850, 9356897577480991804414096100808056657074475)
(111636055166885259237509337117180273290/14682411241, -487823901808888395671909866596340424232768694211466813175/1779082452483211)
(7567536049719797112070653957751397690/696062689, -1255893493734654482148916178853135951569451556729122475/18364221923887)
(-97462007226053929775323704150550/43681, 14085873236445711082925483864645231889476010727675/9129329)
(16464504132790559603714823050, -1271035359109973311611309299789989307521525)
(4587963836280296725878930746984890/405769, -53763405105089402627347290803372081026454573806025/258474853)
(79750152700455655309422906650, 22059798716647636862887589533559461058118875)
(4903285290563186052126990482, -714282345122020177351851941439378400394509)
(-47805618637235929843418921064145/3844, -434791035648061535967022392061755420303970489625/238328)
(2757361066936970844205181343432169323911570/203910488956729, -1952180001440922695657417352439116005792312164428313960026377735/2911785299096649106067)

Commentary

Found in this search by specializing Noam D. Elkies's rank-17 elliptic K3 family (arXiv:2608.25406) at t=-17950/7809.

last edited by 7fff-zip at 2026-08-29 12:32:49 UTC · history