curve #3585

y² = x³ + x² − 349607318480x + 79525435374086100

a-invariants: [0, 1, 0, -349607318480, 79525435374086100]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 114682154160
discriminant (Δ): 2671097302682984107634637519360000
Faltings height: 5.3607
naive height: 91.3538
primes of bad reduction: 2, 3, 5, 7, 29, 37, 113, 563
regulator: 211.3095341324471874534731725118502751552121978
submitted by: Daniel Błażewicz
submitted at: 2026-09-28 21:46:10 UTC
last updated: 2026-09-28 21:46:10 UTC

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

(138628, 183641514)
(328090, 11810400)
(332290, 6699000)
(1436610, 1594435080)

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)=(1998, 18270), with v-u=16272=113*12^2. PARI/GP gives exact rank 4 and torsion C2 x C4.

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