curve #2812

y² + xy = x³ − 61209184x + 184309079759

a-invariants: [1, 0, 0, -61209184, 184309079759]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 61208589
discriminant (Δ): 963031859491327872129
Faltings height: 3.1122
naive height: 65.4030
primes of bad reduction: 3, 13, 23, 29, 181
regulator: 59.96368580584818674549371459219365298144949254
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:23:33 UTC
last updated: 2026-09-26 20:23:33 UTC

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

(195677, -86588236)
(30902, -5288251)
(358906/25, -188999269/125)
(4655, -18298)

Commentary

2-isogenous to curve#850 via the rational 2-torsion point x=-2248. 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 20:23:33 UTC · history