curve #3010

y² + xy = x³ − 25629257095886449788387798512582230x + 1579254561855589758405784950165972094815270292484900

a-invariants: [1, 0, 0, -25629257095886449788387798512582230, 1579254561855589758405784950165972094815270292484900]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 152125511168293484465581800363930
discriminant (Δ): 109088779417669092878744295651414949219750559446567665337018377719787528386054123318795712000
Faltings height: 18.0456
naive height: 249.3007
primes of bad reduction: 2, 3, 5, 11, 13, 17, 23, 29, 31, 73, 139, 167, 227, 251, 499, 2093890919
regulator: 415273884.484392561728754381669849886881962947182633557
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:33:10 UTC
last updated: 2026-09-26 20:33:10 UTC

Witness: 8 independent points log in to add more points →

(155388413688971783516/1681, -336311789269181201837073638/68921)
(92459587550434982, -16205560242171352921474)
(157355170324156420, 37981241156453790338792770)
(256273178549112340220/1681, 2402654841631484192673797290810/68921)
(37457614190233003683278120/405257161, 1856937715316597898332265677467070/8158231908091)
(6517855317671712108739009095775/70555732864516, 15600775153640319672872282126549450601675145/592650234905786812936)
(161401710795746036, 40586276713985948541717242)
(216679494873321769879700/2343961, 24153180046901876908066510085870/3588604291)

Commentary

2-isogenous image of D-z6old678-013, rational kernel x=-46214398621289140. Source: https://dujella.github.io/tors/z6old678.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:33:11 UTC · history