curve #3584

y² = x³ + x² − 136147648080x + 19331747261718228

a-invariants: [0, 1, 0, -136147648080, 19331747261718228]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 12067267920
discriminant (Δ): 67820678763755950186411107225600
Faltings height: 5.0965
naive height: 88.5246
primes of bad reduction: 2, 3, 5, 71, 73, 89, 109
regulator: 9242.847619273834077397377037014521771120668228
submitted by: Daniel Błażewicz
submitted at: 2026-09-28 21:40:06 UTC
last updated: 2026-09-28 21:40:06 UTC

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

(1033902843, 33244481732112)
(-2357526/49, 55062790560/343)
(5192447883/27889, 97393728124440/4657463)
(34491557841/160000, 18330671582361/64000000)

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)=(-14418, 1278), with v-u=15696=109*12^2. PARI/GP gives exact rank 4 and torsion C2 x C4.

last edited by Daniel Błażewicz at 2026-09-28 21:43:22 UTC · history