curve #2540

y² + xy = x³ − 411937506x − 3218101426764

a-invariants: [1, 0, 0, -411937506, -3218101426764]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 6630
discriminant (Δ): 441370202660156250000
Faltings height: 3.4276
naive height: 71.1227
primes of bad reduction: 2, 3, 5, 13, 17
regulator: 7.602028319163792105439695578195627795514
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:07:58 UTC
last updated: 2026-09-26 20:07:58 UTC

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

(-12759446/1089, 200140234/35937)

Commentary

Exact computational full 2- and 3-torsion family at (p,q)=(-13,4); Velu quotient by rational order-3 subgroup. The mapped rank >=1 witness is non-torsion by the original exact 2-descent Kummer certificate and finite isogeny kernel. Conductor N=6630; local exponents {'2': 1, '3': 1, '5': 1, '13': 1, '17': 1}.

last edited by Natanael Wildner Fraga at 2026-09-26 20:07:58 UTC · history