curve #2665

y² + xy + y = x³ − 11784345833x + 486582464844056

a-invariants: [1, 0, 1, -11784345833, 486582464844056]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 31141110
discriminant (Δ): 2454292447815656063338785562500
Faltings height: 4.6377
naive height: 81.1837
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 61
regulator: 4.34460678347978652192451705344753661387223
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:16:38 UTC
last updated: 2026-09-26 20:16:38 UTC

Witness: 2 independent points log in to add more points →

(-124548, 4783399)
(-102435, 24928342)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-22, q=13; A=(3q-p)(p+q)^3=-44469, B=-(p+3q)(p-q)^3=728875. 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 20:16:38 UTC · history