curve #5008

y² = x³ − 529x

a-invariants: [0, 0, 0, -529, 0]
c-invariants: [25392, 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): 16928
discriminant (Δ): 9474296896
Faltings height: 0.6038
naive height: 30.4266
primes of bad reduction: 2, 23
regulator: 10.36787921155264717663573590490179183622
submitted by: David Renshaw
submitted at: 2026-10-21 01:46:39 UTC
last updated: 2026-10-21 01:46:39 UTC

Congruent-number curve y^2 = x^3 - n^2 x with n = 23. 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 01:46:39 UTC · history