curve #3012

y² + xy = x³ − 3414223412309527657925117867276x + 2048517896919356527569649754303110008379265680

a-invariants: [1, 0, 0, -3414223412309527657925117867276, 2048517896919356527569649754303110008379265680]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 1991379159914908277187907157210
discriminant (Δ): 734301549282852131907312401457717474529939170364457462102069294774781956830985263452363161600
Faltings height: 16.5405
naive height: 222.5301
primes of bad reduction: 2, 5, 11, 13, 19, 23, 31, 43, 53, 61, 73, 97, 103, 223, 271, 1987, 8443
regulator: 7033607.32397847601202648198731196589829300140211528811
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:33:14 UTC
last updated: 2026-09-26 20:33:14 UTC

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

(2786059168719804, 119004330069550905194660)
(581806489926768, 16094761323775942519516)
(-26494341925818512908520/37442161, 14688241181936141211623784056521900/229108583159)
(424028777166828, 26019789637417966394276)
(152388600176653639992/76729, -1183668701078723418740783594812/21253933)
(473654344347928, 23186628611991338815796)
(449432353432344, 24593413997572330000180)
(13022828720789726156312/2879809, -1374327401098457188583615573472332/4887035873)

Commentary

2-isogenous image of D-z6old678-060, rational kernel x=-11158741786780865/4. 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 20:33:14 UTC · history