curve #3570

y² = x³ − x² − 62775334402240x + 190881907114676878912

a-invariants: [0, -1, 0, -62775334402240, 190881907114676878912]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 10674865284720
discriminant (Δ): 92105432614722265086892390233170434560000
Faltings height: 6.7245
naive height: 106.9254
primes of bad reduction: 2, 3, 5, 7, 13, 37, 41, 43, 59, 127
regulator: 2438.884293128970803398924407952527388297491092
submitted by: Daniel Błażewicz
submitted at: 2026-09-28 19:19:22 UTC
last updated: 2026-09-28 19:19:22 UTC

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

(62848297/16, 141755403699/64)
(100702834/121, 15704293591350/1331)
(691169826/169, 3509493787450/2197)
(2158912576/529, 19849276235880/12167)

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)=(1270,7462), with v-u=6192=43*12^2. PARI/GP gives exact rank 4 and torsion C2 x C4.

last edited by Daniel Błażewicz at 2026-09-28 19:19:23 UTC · history