curve #5026

y² = x³ − 21799561x

a-invariants: [0, 0, 0, -21799561, 0]
c-invariants: [1046378928, 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): 697585952
discriminant (Δ): 663014791777527602910784
Faltings height: 3.2604
naive height: 62.3058
primes of bad reduction: 2, 7, 23, 29
regulator: 131.74903547417805034340133676023535187610993
submitted by: David Renshaw
submitted at: 2026-10-21 01:48:16 UTC
last updated: 2026-10-21 01:48:16 UTC

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

(10469, 958740)
(14812, 1710786)
(21025/4, 1400265/8)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 4669. 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 01:48:17 UTC · history