curve #2684

y² + xy + y = x³ + x² − 118188x − 2939844

a-invariants: [1, 1, 1, -118188, -2939844]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 608475
discriminant (Δ): 102047903187890625
Faltings height: 1.9539
naive height: 46.6537
primes of bad reduction: 3, 5, 7, 19, 61
regulator: 12.328537419931489130596568595223112460534475
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:17:37 UTC
last updated: 2026-09-26 20:17:37 UTC

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

(-221, 3638)
(-329, 848)
(416, -4692)

Commentary

2-isogenous to curve#1528 via the rational 2-torsion point x=165. 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 20:17:37 UTC · history