curve #742

y² + xy = x³ − 2574544395995602431572694772414793335x + 1590142783521180322672733098620485660577223586022749225

a-invariants: [1, 0, 0, -2574544395995602431572694772414793335, 1590142783521180322672733098620485660577223586022749225]
rank (lower bound): ≥ 21
torsion subgroup: trivial
conductor (N): 578428639984362476094839549654682846384775913940539978193809925490248572670
naive height: 263.1300
Faltings height: 19.6185
discriminant (Δ): -188276824185236699074697649198046544797560571814523252727185629091630037535751953678619982561738733568000000
primes of bad reduction: 2, 3, 5, 13, 17, 31, 73, 83, 89, 109, 211, 229, 8237, 157159319, 4481778631002221601761, 10046866790448265616569
regulator: 89612332731499927945.930879657378311461979382236353148117973442872315021250559823
submitted by: Valery Asiryan
submitted at: 2026-09-16 18:15:11 UTC
last updated: 2026-09-16 18:15:11 UTC

Witness: 21 independent points log in to add more points →

(-1116194618116980740, -1753048542612704384302902875)
(3901133798208439836605/21904, -3457127812655114447796930752324245/3241792)
(878023134699582230, -80764463198966039221039435)
(925370031426195742, -11827270663881164879007899)
(927582477435021170, -11876885897491734411403135)
(934756649775495830, -18237373299467504763464395)
(942529381285708070, -29427686006039397366296635)
(980558949703449620, -91943457070211226178812235)
(4084690114456231775/4, -1288985403905662391374835195/8)
(1035165746259221270, -185238299681285550869858635)
(1104753899596237670, -306979094114644274237750635)
(1658474571182554070, -1371868631570175632106986635)
(763461662713934141350/441, -14110878077486290471360203848815/9261)
(7557393901746312062, -20341283714330006674225398619)
(32956323755350021270, -188974181066107220389210962635)
(295520153816963777270, -5080123473497357112408566966635)
(6610176389941686085510/9, -537425831956531757999216713628585/27)
(849654256744809110, -126673141070283577607190475)
(926088525148392070, -11716793797166084288585435)
(50151813630880722550, -354985260154762044042190975115)
(924844521998451350, -11983394182984115458099915)

Commentary

Specialization T=17/106 of the family identified by wgxli in the commentary to curve #302. The points were obtained by specializing reconstructed generic sections and running minimized quartic searches. The 21 listed points are certified independent by exact quadratic-character certificates.

last edited by Valery Asiryan at 2026-09-16 18:15:11 UTC · history