curve #2451

y² + xy = x³ − 7512579080x + 236240568590400

a-invariants: [1, 0, 0, -7512579080, 236240568590400]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 40750710
discriminant (Δ): 3026204364360991776266304000000
Faltings height: 4.5990
naive height: 79.8331
primes of bad reduction: 2, 3, 5, 7, 11, 13, 23, 59
regulator: 1.814608121928480412424725920048407299929
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:02:24 UTC
last updated: 2026-09-26 20:02:24 UTC

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

(-68240, 20797720)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-23, q=12; A=(3q-p)(p+q)^3=-78529, B=-(p+3q)(p-q)^3=557375. 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 20:02:24 UTC · history