curve #5024

y² = x³ − 12264004x

a-invariants: [0, 0, 0, -12264004, 0]
c-invariants: [588672192, 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): 196224064
discriminant (Δ): 118052944679228203012096
Faltings height: 3.1166
naive height: 60.5801
primes of bad reduction: 2, 17, 103
regulator: 282.43333606339253962281593416439892501326912
submitted by: David Renshaw
submitted at: 2026-10-21 01:48:09 UTC
last updated: 2026-10-21 01:48:09 UTC

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

(3698, 72240)
(5202, 277440)
(449650, 301507920)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 3502. 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 01:48:09 UTC · history