curve #2682

y² + xy = x³ − 3099197884060x + 2099891075269145600

a-invariants: [1, 0, 0, -3099197884060, 2099891075269145600]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 2615851410
discriminant (Δ): 221542081226883833711672772506250000
Faltings height: 5.8373
naive height: 97.9001
primes of bad reduction: 2, 3, 5, 19, 23, 61, 3271
regulator: 60.661125031086576150044314826802954701685397
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:17:33 UTC
last updated: 2026-09-26 20:17:33 UTC

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

(57658430/49, 98230126520/343)
(57350000/49, -94668867880/343)
(942993770/841, -4548389505290/24389)

Commentary

2-isogenous to curve#1567 via the rational 2-torsion point x=-505032. 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 20:17:33 UTC · history