curve #5666

y² + y = x³ − 1460216329829085984303505763887x + 682209983973592675601441107777167588238113756

a-invariants: [0, 0, 1, -1460216329829085984303505763887, 682209983973592675601441107777167588238113756]
c-invariants: [70090383831796127246568276666576, -589429426153184071719645117119472796237730285400]
rank (lower bound): ≥ 22
torsion subgroup: trivial
conductor (N): 583945751773930214642842522548556249047256451031326349952152668504666354219282831713
discriminant (Δ): -1792065862107248470345487631866561335694883897261978153431011754999954032466369108698540283
Faltings height: 16.1637
naive height: 219.9910
primes of bad reduction: 7, 17, 37, 41, 563723, 39486137, 145321395146795301244789402693108389357264661341034471363305966481
regulator: 140787918997245571109305.3242597793791848312064927554224898431654765146258775541867
submitted by: apkorte
submitted at: 2026-10-24 02:40:09 UTC
last updated: 2026-10-24 02:40:09 UTC

Witness: 22 independent points log in to add more points →

(144223249040607, -21785608414774053840666)
(761832742348841, -3453811443751848676011)
(-631195951455979, -36775248404812088601291)
(-112581433494125/9, -714594643453179132417406/27)
(-1152871127501907, -28867915908001569919659)
(1274284935438460583/11449, 27963877605840724207692847762/1225043)
(442070749758269, 11094296586164842128746)
(-1339058024096238, 15378265408069333264365)
(819330593815064, -5985720277249250725989)
(1462040331612250, 40896412699734428832602)
(664206443926811, -2313650359206731001895)
(950507964516882, 12369873968380675529709)
(673061157198080003137/1127844, -5785098220244258146195464982877/1197770328)
(728221331851071, -2242692465368166783591)
(57034444701897254/121, -13219855800312250403289189/1331)
(738450560494416, -2568330882493505330461)
(675088913957876, 2025475993839511631279)
(581641483938000, 5446179340732698001052)
(1021511139004344991/1225, 285729914908640659561610211/42875)
(211782720722947291/361, 35950610700838138639793666/6859)
(-727993343511873, -36870313038762570483610)
(32413652612246, 25197481905099219144390)

Commentary

Specialization at T = 81/37 of the rank-17 fibration of X1008 given in curve#3589. Found by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-24 02:40:09 UTC · history