curve #2510

y² + xy = x³ − x² − 82132479x + 3449491353

a-invariants: [1, -1, 0, -82132479, 3449491353]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 296010
discriminant (Δ): 35453780808504447380062500
Faltings height: 3.5920
naive height: 66.2851
primes of bad reduction: 2, 3, 5, 11, 13, 23
regulator: 3.487515320223940045285854548901448724502
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:06:05 UTC
last updated: 2026-09-26 20:06:05 UTC

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

(-8261, 347831)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-18, q=5; A=(3q-p)(p+q)^3=-72501, B=-(p+3q)(p-q)^3=-36501. 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 20:06:05 UTC · history