curve #2685

y² + xy = x³ − 1867502265314x + 878986611300042564

a-invariants: [1, 0, 0, -1867502265314, 878986611300042564]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 2434962642
discriminant (Δ): 83063377742955889354772628074406084864
Faltings height: 6.0081
naive height: 96.3805
primes of bad reduction: 2, 3, 7, 31, 67, 103, 271
regulator: 141.10696100860225752479929841433587960420589
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:17:41 UTC
last updated: 2026-09-26 20:17:41 UTC

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

(1219468, -644893274)
(34463699236/19321, -4823996476116422/2685619)
(3561525196/1849, -167389003036046/79507)

Commentary

2-isogenous to curve#1566 via the rational 2-torsion point x=-4557345/4. 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 20:17:41 UTC · history