curve #3000

y² + xy = x³ − 268603569295593459578064712908330x + 1693229299441056049976534667502758242071692532900

a-invariants: [1, 0, 0, -268603569295593459578064712908330, 1693229299441056049976534667502758242071692532900]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 36066743023784036920315345816430
discriminant (Δ): 1712354480578621378073195764543265928593179857329667367025709630306054586089784169803808768000000
Faltings height: 17.3956
naive height: 235.6260
primes of bad reduction: 2, 5, 7, 11, 13, 19, 29, 37, 47, 61, 113, 131, 181, 191, 211, 1543, 369983
regulator: 8400141.04818399343055091315567196986395242615651457115
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:32:43 UTC
last updated: 2026-09-26 20:32:43 UTC

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

(11419439062521560, 339209428614138189111970)
(783071783150077492/81, -4303185666791220981949546/729)
(216850812423210330260/22201, 128332829667412943213851153230/3307949)
(3570824800484402310/361, -441248703802786220335752770/6859)
(13813796884486109457940/18601969, 98070634559813113801145402609359190/80230292297)
(26426599304749082304548340/4299031489, 147648202657312726721532142974255847610/281874597639263)
(-163527623527207310/9, 20476278346300752393101740/27)
(23236510292447055220/2401, -1484398647695682069653785370/117649)

Commentary

2-isogenous image of D-z6old678-071, rational kernel x=-74068089074960289/4. 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 20:32:43 UTC · history