curve #5014

y² = x³ − 19044x

a-invariants: [0, 0, 0, -19044, 0]
c-invariants: [914112, 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): 304704
discriminant (Δ): 442032795979776
Faltings height: 1.4997
naive height: 41.1771
primes of bad reduction: 2, 3, 23
regulator: 9.62301306493471118433887270103159659143917
submitted by: David Renshaw
submitted at: 2026-10-21 01:47:17 UTC
last updated: 2026-10-21 01:47:17 UTC

Congruent-number curve y^2 = x^3 - n^2 x with n = 138. 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 01:47:17 UTC · history