curve #3580

y² = x³ − x² − 7244367045024x + 7504666157066102784

a-invariants: [0, -1, 0, -7244367045024, 7504666157066102784]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 212869875504
discriminant (Δ): 1965929191474632918294232537907871744
Faltings height: 6.0387
naive height: 100.4473
primes of bad reduction: 2, 3, 11, 13, 113, 127, 2161
regulator: 3815.367706124317997014906808707886899821935761
submitted by: Daniel Błażewicz
submitted at: 2026-09-28 20:32:12 UTC
last updated: 2026-09-28 20:32:12 UTC

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

(1568240, 25503632)
(5455461/4, 3212857557/8)
(479596720/289, 1128968830704/4913)
(17316494265/17161, 2485220942837988/2248091)

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)=(1016,17288), 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 20:32:23 UTC · history