curve #2822

y² = x³ − x² + 1386815x + 158165761

a-invariants: [0, -1, 0, 1386815, 158165761]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
conductor (N): 1147880256
discriminant (Δ): -181571032719477669888
Faltings height: 2.5765
naive height: 54.0412
primes of bad reduction: 2, 3, 17, 137, 151
regulator: 116.612655249416327038751471179870241825890192215
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:23:59 UTC
last updated: 2026-09-26 20:23:59 UTC

Witness: 5 independent points log in to add more points →

(2488, 137853)
(193, -20808)
(31, 14184)
(-163/4, 80631/8)
(-96, -4913)

Commentary

2-isogenous to curve#1269 via the rational 2-torsion point x=57. 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 20:23:59 UTC · history