curve #5011

y² = x³ − 1681x

a-invariants: [0, 0, 0, -1681, 0]
c-invariants: [80688, 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): 53792
discriminant (Δ): 304006671424
Faltings height: 0.8928
naive height: 33.8950
primes of bad reduction: 2, 41
regulator: 20.0624975035754025769379354384241124029425
submitted by: David Renshaw
submitted at: 2026-10-21 01:47:03 UTC
last updated: 2026-10-21 01:47:03 UTC

Congruent-number curve y^2 = x^3 - n^2 x with n = 41. 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 01:47:03 UTC · history