curve #2768

y² = x³ − 7070700x + 4436606000

a-invariants: [0, 0, 0, -7070700, 4436606000]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 132033600
discriminant (Δ): 14120625938457600000000
Faltings height: 2.9508
naive height: 58.9280
primes of bad reduction: 2, 3, 5, 53, 173
regulator: 66.78138393338677327602095297217607635160535507
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:21:34 UTC
last updated: 2026-09-26 20:21:34 UTC

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

(9305/4, -192125/8)
(-10199/4, -613899/8)
(40420, -8109000)
(10045, -973125)

Commentary

2-isogenous to curve#1724 via the rational 2-torsion point x=5860. 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 20:21:34 UTC · history