curve #3566

y² = x³ − x² − 380989052195360x + 2861795706513140889600

a-invariants: [0, -1, 0, -380989052195360, 2861795706513140889600]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 3250497013680
discriminant (Δ): 1275165844926449727808127025206841600000000
Faltings height: 7.0757
naive height: 112.3350
primes of bad reduction: 2, 3, 5, 7, 29, 37, 71, 109, 233
regulator: 68593.2459687626429232906235691166725351656011182
submitted by: jesper-petersen
submitted at: 2026-09-28 17:44:24 UTC
last updated: 2026-09-28 17:44:24 UTC

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

(11131802, 347236722)
(-47655300097680/2217121, 110506559938170270000/3301293169)
(96966505/9, 74696217400/27)
(654597680/49, 4287766935600/343)
(55517508080/5329, 1880467759371600/389017)

Commentary

Found in a targeted low-conductor search in the C2 x C4 family y^2 = x(x+u^2)(x+v^2), using low-squarefree-kernel loci u±v=d*w^2. Here (u,v)=(497,5825), with v-u=5328=37*12^2. PARI/GP gives exact rank 5 and torsion C2 x C4.

last edited by jesper-petersen at 2026-09-28 17:44:24 UTC · history