curve #5025

y² = x³ − 17631601x

a-invariants: [0, 0, 0, -17631601, 0]
c-invariants: [846316848, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 564211232
discriminant (Δ): 350796475913120292787264
Faltings height: 3.2073
naive height: 61.6692
primes of bad reduction: 2, 13, 17, 19
regulator: 131.95301266788511381750334132107649401032811
submitted by: David Renshaw
submitted at: 2026-10-21 01:48:13 UTC
last updated: 2026-10-21 01:48:13 UTC

Witness: 3 independent points log in to add more points →

(-3249, 151620)
(4225, 30420)
(25857, 4102644)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 4199. Found in an exhaustive ellrank scan of squarefree n <= 20000; one of the smallest n of rank 3.

last edited by David Renshaw at 2026-10-21 01:48:13 UTC · history