curve #5005

y² = x³ − 225x

a-invariants: [0, 0, 0, -225, 0]
c-invariants: [10800, 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): 7200
discriminant (Δ): 729000000
Faltings height: 0.3901
naive height: 27.8619
primes of bad reduction: 2, 3, 5
regulator: 1.491344811919626301861768688987684568791
submitted by: David Renshaw
submitted at: 2026-10-21 01:46:27 UTC
last updated: 2026-10-21 01:46:27 UTC

Congruent-number curve y^2 = x^3 - n^2 x with n = 15. 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 01:46:27 UTC · history