curve #736

y² + xy + y = x³ − x² − 62330715385053591956537219402x + 5994456478488911103022452992502858001259401

a-invariants: [1, -1, 1, -62330715385053591956537219402, 5994456478488911103022452992502858001259401]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 528231522282148305732318171943623605900920769594482376831197710
naive height: 210.5220
Faltings height: 15.3089
discriminant (Δ): -24895544437583552594644825153545076278982533771778054578056326482615597253926912000000
primes of bad reduction: 2, 3, 5, 7, 17, 19, 43, 59, 23011943137, 25265118093587, 1759892540399429283515554272293
regulator: 160528730390667013.1537921800732385331731274803042248452855625597056274461987
submitted by: Valery Asiryan
submitted at: 2026-09-16 17:21:34 UTC
last updated: 2026-09-16 17:21:34 UTC

Witness: 19 independent points log in to add more points →

(-160763371664111, -3443841573927531887881)
(42721767645841, -1846497177738479971081)
(108718858323769, -709201263202549497241)
(140774590434241, -98356193267497121881)
(141673259686351, -86192418544744510951)
(41021011752661999/289, -407881460392446186720503/4913)
(810299143053652051/5476, -42905988999578653555163999/405224)
(152283068712571, -184413969899801402641)
(159979187135461, -342404924486269771141)
(77647912115897341/441, -6404459071744715324981261/9261)
(204427092487241, -1339950442381527154361)
(380301010985155, -6106760680037678842273)
(3459095905080691, -202927724403572519020531)
(4194869565495001, -271221966834037704844801)
(6071322492482479, -472675706603494827014569)
(27597984435565489, -4584567168898733491723081)
(98018114961523601, -30687259965939258848536201)
(143842122625601, -69624785948937034201)
(143936585219281, -69476557307295291721)

Commentary

Specialization T=4/3 of the family identified by wgxli in the commentary to curve #302. Starting from 17 reconstructed generic sections, minimized quartic searches produced a set of 19 points certified independent by exact quadratic-character certificates.

last edited by Valery Asiryan at 2026-09-16 17:21:34 UTC · history