curve #5021

y² = x³ − 51076x

a-invariants: [0, 0, 0, -51076, 0]
c-invariants: [2451648, 0]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 817216
discriminant (Δ): 8527674378686464
Faltings height: 1.7463
naive height: 44.1368
primes of bad reduction: 2, 113
regulator: 34.6187300286877206249893575917630697439725
submitted by: David Renshaw
submitted at: 2026-10-21 01:47:46 UTC
last updated: 2026-10-21 01:47:46 UTC

Congruent-number curve y^2 = x^3 - n^2 x with n = 226. Found in an exhaustive ellrank scan of squarefree n <= 20000; one of the smallest n of rank 2.

last edited by David Renshaw at 2026-10-21 01:47:46 UTC · history