curve #2756

y² + xy = x³ − 49734998132196451150x + 135000265917508085623660512500

a-invariants: [1, 0, 0, -49734998132196451150, 135000265917508085623660512500]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 566645048606910
discriminant (Δ): 241064784414476125389168415624251443098654812149076810000
Faltings height: 9.9462
naive height: 147.6733
primes of bad reduction: 2, 3, 5, 7, 13, 19, 37, 97, 101, 30137
regulator: 2897.894518848233385033795135185355965011404574
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:21:00 UTC
last updated: 2026-09-26 20:21:00 UTC

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

(10391691380, -860435138847130)
(36130041541675/2916, 187699248584411129225/157464)
(68567231810338/15129, -96449500419254153408/1860867)
(264421044093786050/33051001, -94849240430500600630295050/190010204749)

Commentary

2-isogenous to curve#1640 via the rational 2-torsion point x=-2029317700. 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 20:21:00 UTC · history