curve #2694

y² + xy = x³ − 10478512673964x + 13045978923941019024

a-invariants: [1, 0, 0, -10478512673964, 13045978923941019024]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 1242248154
discriminant (Δ): 108728054354148105879321146933670641664
Faltings height: 6.2254
naive height: 101.5546
primes of bad reduction: 2, 3, 7, 73, 97, 4177
regulator: 68.137337013279127840793340406779038884814912
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:18:03 UTC
last updated: 2026-09-26 20:18:03 UTC

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

(328684696/9, -5936782872292/27)
(21581129592/10609, -421637273266068/1092727)
(438486763320/229441, -2344552489831572/109902239)

Commentary

2-isogenous to curve#1564 via the rational 2-torsion point x=-14618305/4. 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 20:18:03 UTC · history