curve #5002

y² = x³ − 49x

a-invariants: [0, 0, 0, -49, 0]
c-invariants: [2352, 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): 1568
discriminant (Δ): 7529536
Faltings height: 0.0090
naive height: 23.2891
primes of bad reduction: 2, 7
regulator: 2.988881507603340609826914941127797535216
submitted by: David Renshaw
submitted at: 2026-10-21 01:46:14 UTC
last updated: 2026-10-21 01:46:14 UTC

Congruent-number curve y^2 = x^3 - n^2 x with n = 7. 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 01:46:14 UTC · history