curve #3571

y² = x³ − x² − 9376814107704x + 11013189122859963264

a-invariants: [0, -1, 0, -9376814107704, 11013189122859963264]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 3333579476304
discriminant (Δ): 367599161688691688170524025270242223104
Faltings height: 6.2563
naive height: 101.2214
primes of bad reduction: 2, 3, 7, 11, 53, 59, 211, 1367
regulator: 6382.872269341372424546570728959478715993348411
submitted by: Daniel Błażewicz
submitted at: 2026-09-28 19:21:40 UTC
last updated: 2026-09-28 19:21:40 UTC

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

(-2781389211/841, 58991603910246/24389)
(11794264113/1024, 1239555396004695/32768)
(16389655781/8836, 38749954968725/830584)
(46744593996/22801, 2194328392183620/3442951)

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)=(1652,9284), with v-u=7632=53*12^2. PARI/GP gives exact rank 4 and torsion C2 x C4.

last edited by Daniel Błażewicz at 2026-09-28 19:21:40 UTC · history