curve #5013

y² = x³ − 18769x

a-invariants: [0, 0, 0, -18769, 0]
c-invariants: [900912, 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): 600608
discriminant (Δ): 423158800038976
Faltings height: 1.4960
naive height: 41.1335
primes of bad reduction: 2, 137
regulator: 84.1583165958681231998266224966512453577973
submitted by: David Renshaw
submitted at: 2026-10-21 01:47:12 UTC
last updated: 2026-10-21 01:47:12 UTC

Congruent-number curve y^2 = x^3 - n^2 x with n = 137. 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 01:47:12 UTC · history