curve #5006

y² = x³ − 441x

a-invariants: [0, 0, 0, -441, 0]
c-invariants: [21168, 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): 14112
discriminant (Δ): 5489031744
Faltings height: 0.5583
naive height: 29.8807
primes of bad reduction: 2, 3, 7
regulator: 1.661457220729774748305437739250849271972
submitted by: David Renshaw
submitted at: 2026-10-21 01:46:31 UTC
last updated: 2026-10-21 01:46:31 UTC

Congruent-number curve y^2 = x^3 - n^2 x with n = 21. 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 01:46:31 UTC · history