curve #2816

y² + xy = x³ − x² + 10542429061476x + 3312840803436975330

a-invariants: [1, -1, 0, 10542429061476, 3312840803436975330]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/6ℤ
conductor (N): 40432009593690
discriminant (Δ): -79730942989378299159250444046708159156250
Faltings height: 6.5374
naive height: 101.5729
primes of bad reduction: 2, 3, 5, 11, 13, 31, 47, 463, 4657
regulator: 27055.8295459624520820240632525769466306093587785
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:23:44 UTC
last updated: 2026-09-26 20:23:44 UTC

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

(159314179/9, 2044758220744/27)
(-3760979/16, 58023331433/64)
(3075000049/625, 206233500342482/15625)
(15235596109/5625, 3033942620527498/421875)
(1738178647519/208849, -2466357051349313179/95443993)

Commentary

2-isogenous to curve#1667 via the rational 2-torsion point x=155688. 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 20:23:44 UTC · history