curve #2469

y² + xy = x³ − x² − 8247861819x + 288187607197233

a-invariants: [1, -1, 0, -8247861819, 288187607197233]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 4823910
discriminant (Δ): 31073030682941202153215062500
Faltings height: 4.4198
naive height: 80.1133
primes of bad reduction: 2, 3, 5, 7, 13, 19, 31
regulator: 5.341180344797384660000792675053895790776
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:03:30 UTC
last updated: 2026-09-26 20:03:30 UTC

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

(59913, 2985426)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-18, q=13; A=(3q-p)(p+q)^3=-7125, B=-(p+3q)(p-q)^3=625611. 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 20:03:30 UTC · history