curve #3473

y² = x³ − x² − 728913292028484x + 5987060757049444629444

a-invariants: [0, -1, 0, -728913292028484, 5987060757049444629444]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 38019384450552
discriminant (Δ): 9301070920556501722516656067077726090938513664
Faltings height: 7.5344
naive height: 114.2813
primes of bad reduction: 2, 3, 37, 47, 73, 97, 103, 1249
regulator: 11891.0701626527169885580441754187722545755216233
submitted by: jesper-petersen
submitted at: 2026-09-27 07:20:35 UTC
last updated: 2026-09-27 07:20:35 UTC

Witness: 5 independent points log in to add more points →

(-21809604, 107286718110)
(8828098, 15497202840)
(39632744, 198372206186)
(-9527640348/361, 567311611376910/6859)
(7684024869/3025, 10720393410010122/166375)

Commentary

Found by a conductor-oriented search in the standard Z/2Z x Z/4Z family y^2 = x(x+u^2)(x+v^2), using the square-sum condition u+v=w^2. Here (u,v)=(3431,7178), with u+v=10609=103^2. The relevant factorization is u=4773, v=23797, v-u=31249, v+u=103^2. PARI/GP 2.17.4 gives exact rank 5 and torsion Z/2Z x Z/4Z. Conductor N=38019384450552.

last edited by jesper-petersen at 2026-09-27 07:20:35 UTC · history