curve #5012

y² = x³ − 4225x

a-invariants: [0, 0, 0, -4225, 0]
c-invariants: [202800, 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): 135200
discriminant (Δ): 4826809000000
Faltings height: 1.1232
naive height: 36.6599
primes of bad reduction: 2, 5, 13
regulator: 9.30711951326421589201333839735396665352298
submitted by: David Renshaw
submitted at: 2026-10-21 01:47:07 UTC
last updated: 2026-10-21 01:47:07 UTC

Congruent-number curve y^2 = x^3 - n^2 x with n = 65. 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 01:47:08 UTC · history