curve #2270

y² + xy + y = x³ − 5284615102394x − 2083357005192081124

a-invariants: [1, 0, 1, -5284615102394, -2083357005192081124]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 4095010830
discriminant (Δ): 7570351750400080079130007456205705763600
Faltings height: 6.3467
naive height: 99.5011
primes of bad reduction: 2, 3, 5, 17, 29, 43, 47, 137
regulator: 5.118865964194567396452970609733797698239
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:51:02 UTC
last updated: 2026-09-26 19:51:02 UTC

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

(2475758, 88607673)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-4, q=47; A=(3q-p)(p+q)^3=11528515, B=-(p+3q)(p-q)^3=18173187. 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 19:51:03 UTC · history