curve #346

y² + xy = x³ − 1650879174392173933380282511725x + 827912259081400825673880665852533513855100625

a-invariants: [1, 0, 0, -1650879174392173933380282511725, 827912259081400825673880665852533513855100625]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 799749988490452666919632124716837519159126114433589737854386523851870
naive height: 220.3781
Faltings height: 16.2442
discriminant (Δ): -8153715222927956302195203530071282231036456187252543215356220806381515287867179840000000000
primes of bad reduction: 2, 3, 5, 13, 23, 31, 4216697847056923, 461847537619525509593513, 1476824848248452589410159
regulator: 401836351919833283356.0279594182246569487376264081251835724201924095533359904
submitted by: Christopher R. Hill
submitted at: 2026-08-25 04:05:00 UTC
last updated: 2026-08-25 04:05:00 UTC

Witness: 19 independent points

(-3069528923984800830/6241, -19227855505229860682582850255/493039)
(2181668163423686850/2209, 1316538885790405221756990225/103823)
(1654529263324849506, -2128195637027110066526334753)
(100176665612807250/121, 7119276619008014161423725/1331)
(704967033346320, -3801360193690192395495)
(636458637770850, 5917123771763849795775)
(5338251532814842020325/853776, -382380799804246101859218127279525/788889024)
(2913460687526768976075/630436, -151743196956997467402900748598025/500566184)
(34893243661712270968050/78659161, 9434082018554113962233171657489475/697628098909)
(3768321088293550050/5329, -1461460543777262270856456825/389017)
(201935356641811906464450/269255281, 15065413389926006232079312179249975/4418209905929)
(847249363339650, 6114588455401318055775)
(-663399875436430, 40387416863224353722255)
(19391526656520306/25, 468810089449981442255139/125)
(20268223487416898130/76729, 430495144312116608458308448275/21253933)
(6624573282867330, -529727055468507227993505)
(651943419976002, 5359859862410393527071)
(-1473296219998800, -7887137543232907029225)
(37303032483977700, -7200487525462795890942225)

Commentary

Found with Rank Hunter v0.7.21; family=Mestre/Fermigier sextuple, generic rank >=11; parameter=57894/65; local status: rigorous rank lower bound >= 19.

last edited by Christopher R. Hill at 2026-08-25 04:05:00 UTC · history