curve #2680

y² + xy + y = x³ − x² − 1828980x − 935917978

a-invariants: [1, -1, 1, -1828980, -935917978]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 684475
discriminant (Δ): 12790507315828515625
Faltings height: 2.4599
naive height: 54.8714
primes of bad reduction: 5, 11, 19, 131
regulator: 14.319064998139118729160737691390388925151563
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:17:26 UTC
last updated: 2026-09-26 20:17:26 UTC

Witness: 3 independent points log in to add more points →

(552499/36, -410670605/216)
(71506/9, -18930335/27)
(4903, -331000)

Commentary

2-isogenous to curve#1530 via the rational 2-torsion point x=349. Independent witness points are the images of that curve's witnesses under 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 20:17:26 UTC · history