curve #5000

y² = x³ − 36x

a-invariants: [0, 0, 0, -36, 0]
c-invariants: [1728, 0]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 576 ☆ record for congruent-number curves, rank ≥ 1
discriminant (Δ): 2985984
Faltings height: -0.0681
naive height: 22.3642
primes of bad reduction: 2, 3
regulator: 0.8886258748396192397941873349806892865718
submitted by: David Renshaw
submitted at: 2026-10-21 01:46:10 UTC
last updated: 2026-10-21 01:46:10 UTC

Congruent-number curve y^2 = x^3 - n^2 x with n = 6. Found in an exhaustive ellrank scan of squarefree n <= 20000; one of the smallest n of rank 1.

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