curve #5019

y² = x³ − 37636x

a-invariants: [0, 0, 0, -37636, 0]
c-invariants: [1806528, 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): 602176
discriminant (Δ): 3411853332189184
Faltings height: 1.6700
naive height: 43.2208
primes of bad reduction: 2, 97
regulator: 38.6439672225802574929556780889395881939794
submitted by: David Renshaw
submitted at: 2026-10-21 01:47:33 UTC
last updated: 2026-10-21 01:47:33 UTC

Congruent-number curve y^2 = x^3 - n^2 x with n = 194. 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 01:47:34 UTC · history