curve #3569

y² = x³ − x² − 1022036704280x + 394380521528489472

a-invariants: [0, -1, 0, -1022036704280, 394380521528489472]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 449224966320
discriminant (Δ): 1133536362512549414586614670188160000
Faltings height: 5.7383
naive height: 94.5721
primes of bad reduction: 2, 3, 5, 13, 19, 29, 43, 59, 103
regulator: 292.1028173662118931812687939939870432449073681
submitted by: Daniel Błażewicz
submitted at: 2026-09-28 19:10:29 UTC
last updated: 2026-09-28 19:10:29 UTC

Witness: 4 independent points log in to add more points →

(438734, 174438810)
(4074574, 7992290150)
(8763752, 25778342304)
(534924896/529, 7645621978800/12167)

Commentary

Inspired by (much better) curve#3566 Specialized curve of the C2 x C4 family y^2 = x(x+u^2)(x+v^2), where u±v=d*w^2. Here (u,v)=(1180,5356), with v-u=4176=29*12^2. PARI/GP gives exact rank 4 and torsion C2 x C4.

last edited by Daniel Błażewicz at 2026-09-28 19:10:29 UTC · history