curve #1675

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y² = x³ − 18495762430561062007231722390971205929652x + 963113455903034236082440808210991507704589363640791844784100

a-invariants

[0, 0, 0, -18495762430561062007231722390971205929652, 963113455903034236082440808210991507704589363640791844784100]

rank (lower bound)

≥ 27

torsion subgroup

trivial

conductor (N)

1161602477284733099564916583583658971435606282203819438732881481369436392722177333585848605232058647736

discriminant (Δ)

4227792112681944497834383245316547304261844306556883806152085575971800185034559251979903500060576440893980772979939091712

★ record for rank ≥ 27

Faltings height

21.9851 ★ record for rank ≥ 27

naive height

289.7687 ★ record for rank ≥ 27

primes of bad reduction

2, 3, 7, 13, 17, 19, 31, 37, 109, 123803, 3095227, 1542476496729119568900981821, 1061086852930873566515781035506779283027688998746731

regulator

39259272806230101708519606176.584530433904773353661870942226847678767295259341402258607223299

submitted by

Ritik Jain

submitted at

2026-09-25 04:08:26 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-25 04:44:58 UTC

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

(338772218391579308100769/6561, 202896798958025394954988547144900969/531441)
(59626767016879816862326/3721, 185909057780165489669270479981368938/226981)
(316673773138367789046, -5182929492200652508833355135038)
(46060658218905898783774750/2798929, -3813363030490627016937335079429991521730/4682608217)
(7995059094359700667353/16, 690753876517770620649063793558371/64)
(91298621873071069242072/961, 7502564102156224902549779629862718/29791)
(12162217547425059381412824/97969, -23357996658439991943128413429739829714/30664297)
(-132348217786223268662, 1045359531745990727810863014514)
(73545524555090384094, -25229805025318912553824170414)
(4058512380331282502034, 258410094232271476387744000896894)
(83474719976194478001, -28967661298110094986196076607)
(-133087491544310327727, 1033143316912469010728627325711)
(23933448241395367225537/144, 2721398150028024824352151893173983/1728)
(625356742507609320313/9, -3128290491898980308510960359429/27)
(38667129115328461457241, 7603441524023700223460764301266683)
(1331068428209168216337/16, 853223305999501726359603253497/64)
(142236187574856940155809722138670724262848/2802064735122449786521, 58404031413557811716987699062712794523148705023670253094481534/148325986897723734373989478106819)
(15064745232093314802552321/315844, 77257088618744404676686377534561637857/177504328)
(-105525574314674407418586/841, 27896968733071085813365428750698826/24389)
(5857559663034860948398/81, 44696628344540301906462798099806/729)
(265184951547705167040025183528/3596520841, 3179511175599297760013324046411895628903546/215686951355611)
(-54597427215609357951375/361, 3763387005776520003099263222635185/6859)
(-10455868866064559966, 1074876566199727100267416424606)
(231163873893697415072373679212556785148146025/3486395208998670539571529, 1094230117564325937398866025326465589872256911614423646171636062515/6509759201471032013509790198081742917)
(16931875285398287782087755076438810575700464/229382130389602497525625, 702170367049912606333024403866178028676642905477450341701256562/109859943898222121747830248916578125)
(-627555710484469700422752670125403189900097914232/19515122444959706999321186209, 3366199882659334144153306750783206434466523662915137423459166010943927246/2726195024354291099409363012162199675142927)
(383829206620384781481159807696730828697861953/5300836108941070225089081, 721842727660038791036844973945719200864757144031482773985778212169/12204403714085799556043084825333079021)

Commentary

Specialization at T = 29/124 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 = 6858T^2 - 2232366T - 123553, p = 143664T^2 - 618920T - 6487056, q = 6170T^2 - 7098T + 7564, D = -806237487T^4 + 76922980194T^3 - 1226802677561T^2 - 689131950466T + 1104882072456, E = 815119825T^4 - 9557405330T^3 - 16096328659T^2 + 9197377220T - 46146438924.

last edited by wgxli at 2026-09-25 05:07:02 UTC · history