curve #2405

y² + xy + y = x³ − x² − 54809332808x + 4767440962086731

a-invariants: [1, -1, 1, -54809332808, 4767440962086731]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 7580430
discriminant (Δ): 718991272616515905668595321960000
Faltings height: 5.0715
naive height: 85.7950
primes of bad reduction: 2, 3, 5, 11, 13, 19, 31
regulator: 6.438694510700982008184660144246878452549
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:59:35 UTC
last updated: 2026-09-26 19:59:35 UTC

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

(-178455, 94244977)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-9, q=22; A=(3q-p)(p+q)^3=164775, B=-(p+3q)(p-q)^3=1698087. 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 19:59:35 UTC · history