curve #5029

y² = x³ − 39513796x

a-invariants: [0, 0, 0, -39513796, 0]
c-invariants: [1896662208, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 632220736
discriminant (Δ): 3948446283755858136469504
Faltings height: 3.4091
naive height: 64.0901
primes of bad reduction: 2, 7, 449
regulator: 261.15443343372725192061560195907582161409807
submitted by: David Renshaw
submitted at: 2026-10-21 01:48:24 UTC
last updated: 2026-10-21 01:48:24 UTC

Witness: 3 independent points log in to add more points →

(-14, 23520)
(7168, 291648)
(45486, 9607920)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 6286. Found in an exhaustive ellrank scan of squarefree n <= 20000; one of the smallest n of rank 3.

last edited by David Renshaw at 2026-10-21 01:48:24 UTC · history