curve #2266

y² + xy + y = x³ − 5687545999868x + 2045213988816845306

a-invariants: [1, 0, 1, -5687545999868, 2045213988816845306]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 1068411630
discriminant (Δ): 9967818821123358773285084760475521000000
Faltings height: 6.3683
naive height: 99.7215
primes of bad reduction: 2, 3, 5, 11, 13, 37, 53, 127
regulator: 8.248083947624020720490962207777232765785
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:50:47 UTC
last updated: 2026-09-26 19:50:47 UTC

Witness: 1 independent point log in to add more points →

(-2518455, 630109207)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-8, q=45; A=(3q-p)(p+q)^3=7243379, B=-(p+3q)(p-q)^3=18907379. The listed points are independent modulo torsion by the exact 2-descent Kummer squareclass map. The rank is a proven lower bound; no upper bound is claimed.

last edited by Natanael Wildner Fraga at 2026-09-26 19:50:47 UTC · history