curve #2459

y² + xy + y = x³ − 51430768954x + 4489314628073756

a-invariants: [1, 0, 1, -51430768954, 4489314628073756]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 41532870
discriminant (Δ): 87484573199903669447550483600
Faltings height: 4.7435
naive height: 85.6041
primes of bad reduction: 2, 3, 5, 17, 31, 37, 71
regulator: 5.613683496443248712297469424548618351771
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:02:53 UTC
last updated: 2026-09-26 20:02:53 UTC

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

(130666, 8981)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-20, q=17; A=(3q-p)(p+q)^3=-1917, B=-(p+3q)(p-q)^3=1570243. 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 20:02:53 UTC · history