curve #2257

y² + xy + y = x³ − 9806065727763x + 11800611512264818906

a-invariants: [1, 0, 1, -9806065727763, 11800611512264818906]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 10760046990
discriminant (Δ): 190284339947399411467094207924427562500
Faltings height: 6.2367
naive height: 101.3557
primes of bad reduction: 2, 3, 5, 7, 11, 23, 31, 47, 139
regulator: 3.015122104499669960767936847849107724620
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:50:13 UTC
last updated: 2026-09-26 19:50:13 UTC

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

(1733657, 103437228)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-46, q=31; A=(3q-p)(p+q)^3=-469125, B=-(p+3q)(p-q)^3=21457051. 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 19:50:13 UTC · history