curve #3586

y² = x³ − x² − 30594886944x + 1658863795083264

a-invariants: [0, -1, 0, -30594886944, 1658863795083264]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 6762560016
discriminant (Δ): 644072796305919355861812317405184
Faltings height: 5.0111
naive height: 84.0459
primes of bad reduction: 2, 3, 17, 53, 271, 577
regulator: 1803.562934026105955288637949107393402806376495
submitted by: Daniel Błażewicz
submitted at: 2026-09-28 21:55:43 UTC
last updated: 2026-09-28 21:55:43 UTC

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

(135728, 2579920)
(-647139/25, 6166294434/125)
(7969109/49, 10763594570/343)
(32845077424/2025, 5952237498007568/91125)

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)=(2168, 4616), with v-u=2448=17*12^2. PARI/GP gives exact rank 4 and torsion C2 x C4.

last edited by Daniel Błażewicz at 2026-09-28 21:56:47 UTC · history