curve #6219

y² + xy + y = x³ − x² − 846155080257685755735695747x + 9579596900619829716541065610781637526019

a-invariants: [1, -1, 1, -846155080257685755735695747, 9579596900619829716541065610781637526019]
c-invariants: [40615443852368916275313395841, -8276771722135350105594145027592095912199361]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/3ℤ
conductor (N): 606467578887950117548856168527290
discriminant (Δ): -871026620573830843601194479393400284371666357109003481900139590911875088384000000
Faltings height: 14.3397
naive height: 197.6441
primes of bad reduction: 2, 3, 5, 11, 13, 17, 19, 31, 37, 137, 139, 163, 251, 145633, 373667527
regulator: 214816122876.0360492063009312486872755544704777736670727377
submitted by: apkorte
submitted at: 2026-10-30 23:57:36 UTC
last updated: 2026-10-30 23:57:36 UTC

Witness: 10 independent points log in to add more points →

(-598404321326161780133/67092481, 70413826415196057033797864137006/549554511871)
(-2398334311057283997/339889, 24429363755750626357412669782/198155287)
(986455198917169/81, 23968228857243437270368/729)
(24774673063525107/841, 2465276929264887718067354/24389)
(34429314707008323/1849, 1330223599563524286602902/79507)
(45491409914040027/961, 8191218007846237360147246/29791)
(60081839081626243/2809, 5298118257836923004676522/148877)
(48538529353970619843/4826809, 484515350513978840218948543978/10604499373)
(61128191629525608963/7166329, 1047713023924741861174579568938/19184262733)
(308037527215659971163/14691889, 1829500907040313401386118250802/56314010537)

Commentary

Specialization at T = 8959/652 of the fibration given in curve#6190. 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 2026-10-30 23:57:36 UTC · history