curve #5007

y² = x³ − 484x

a-invariants: [0, 0, 0, -484, 0]
c-invariants: [23232, 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): 7744
discriminant (Δ): 7256313856
Faltings height: 0.5816
naive height: 30.1599
primes of bad reduction: 2, 11
regulator: 4.253144970389411834006507941477202214974
submitted by: David Renshaw
submitted at: 2026-10-21 01:46:35 UTC
last updated: 2026-10-21 01:46:35 UTC

Congruent-number curve y^2 = x^3 - n^2 x with n = 22. 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 01:46:35 UTC · history