curve #3583

y² = x³ − x² − 680773488120x + 216093210025852800

a-invariants: [0, -1, 0, -680773488120, 216093210025852800]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 446214180720
discriminant (Δ): 19648422355159062563798332534809600
Faltings height: 5.5272
naive height: 93.3531
primes of bad reduction: 2, 3, 5, 7, 11, 19, 89, 109, 131
regulator: 414.7477977540629712075522245910702687978318286
submitted by: Daniel Błażewicz
submitted at: 2026-09-28 21:29:18 UTC
last updated: 2026-09-28 21:29:18 UTC

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

(673692, 251442972)
(1234296, 1120825200)
(2030727470/3721, 19136745071950/226981)
(5658927864/10609, 75155217009840/1092727)

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)=(1572, 14388), with v-u=12816=89*12^2. PARI/GP gives exact rank 4 and torsion C2 x C4.

last edited by Daniel Błażewicz at 2026-09-28 21:30:54 UTC · history