curve #2660

y² + xy = x³ − 66358900x + 206788250000

a-invariants: [1, 0, 0, -66358900, 206788250000]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 681870
discriminant (Δ): 227641658827636236960000
Faltings height: 3.3146
naive height: 65.6454
primes of bad reduction: 2, 3, 5, 7, 17, 191
regulator: 2.69524188279579459181578435925299618507240
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:16:15 UTC
last updated: 2026-09-26 20:16:15 UTC

Witness: 2 independent points log in to add more points →

(177872, -75028948)
(55540, -12983300)

Commentary

2-isogenous to curve#1188 via the rational 2-torsion point x=-2200. 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 20:16:15 UTC · history