curve #5031

y² = x³ − 52272900x

a-invariants: [0, 0, 0, -52272900, 0]
c-invariants: [2509099200, 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): 836366400
discriminant (Δ): 9141337800449695296000000
Faltings height: 3.4790
naive height: 64.9296
primes of bad reduction: 2, 3, 5, 241
regulator: 112.25268135038902071197769300274727011506026
submitted by: David Renshaw
submitted at: 2026-10-21 01:48:32 UTC
last updated: 2026-10-21 01:48:32 UTC

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

(-3600, 376200)
(-30, 39600)
(7350, 113400)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 7230. 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 01:48:32 UTC · history