curve #5004

y² = x³ − 196x

a-invariants: [0, 0, 0, -196, 0]
c-invariants: [9408, 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): 3136
discriminant (Δ): 481890304
Faltings height: 0.3556
naive height: 27.4479
primes of bad reduction: 2, 7
regulator: 2.134120849044754020051650801945664635106
submitted by: David Renshaw
submitted at: 2026-10-21 01:46:22 UTC
last updated: 2026-10-21 01:46:22 UTC

Congruent-number curve y^2 = x^3 - n^2 x with n = 14. 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 01:46:23 UTC · history