curve #2287

y² + xy = x³ − x² − 2178016639209x + 1224016524703805913

a-invariants: [1, -1, 0, -2178016639209, 1224016524703805913]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 102612510
discriminant (Δ): 14017823277922392364866430977539062500
Faltings height: 5.9381
naive height: 96.8419
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 67
regulator: 4.640801069835395959889957709085740590812
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:52:11 UTC
last updated: 2026-09-26 19:52:11 UTC

Witness: 1 independent point log in to add more points →

(778912, 9696389)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-42, q=25; A=(3q-p)(p+q)^3=-574821, B=-(p+3q)(p-q)^3=9925179. The listed points are independent modulo torsion by the exact 2-descent Kummer squareclass map. The rank is a proven lower bound; no upper bound is claimed.

last edited by Natanael Wildner Fraga at 2026-09-26 19:52:11 UTC · history