curve #5023

y² = x³ − 7695076x

a-invariants: [0, 0, 0, -7695076, 0]
c-invariants: [369363648, 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): 123121216
discriminant (Δ): 29162094596994521534464
Faltings height: 3.0001
naive height: 59.1819
primes of bad reduction: 2, 19, 73
regulator: 160.25221413983330551971673415765277025362115
submitted by: David Renshaw
submitted at: 2026-10-21 01:48:05 UTC
last updated: 2026-10-21 01:48:05 UTC

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

(-475, 59565)
(3724, 151620)
(-31958/49, 23618064/343)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 2774. 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 01:48:05 UTC · history