curve #2749

y² + xy + y = x³ − x² − 61924856x + 187325279826

a-invariants: [1, -1, 1, -61924856, 187325279826]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 2042704917
discriminant (Δ): 40896077747150509489089
Faltings height: 3.2376
naive height: 65.4379
primes of bad reduction: 3, 11, 31, 37, 17989
regulator: 156.5276683242142980618951557109489556792343123
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:20:36 UTC
last updated: 2026-09-26 20:20:36 UTC

Witness: 4 independent points log in to add more points →

(157503/4, -61501361/8)
(115877/16, 21815631/64)
(21117584/4489, -3771646821/300763)
(583828/9, -443827897/27)

Commentary

2-isogenous to curve#1510 via the rational 2-torsion point x=-2203. 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 20:20:36 UTC · history