curve #2227

y² + xy + y = x³ − 66912931789579x + 210642367183352068706

a-invariants: [1, 0, 1, -66912931789579, 210642367183352068706]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 233395890
discriminant (Δ): 5956057222412798076478078846113521122500
Faltings height: 6.6360
naive height: 107.1168
primes of bad reduction: 2, 3, 5, 7, 13, 17, 47, 107
regulator: 7.261252017459820525355376540043125266918
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:48:21 UTC
last updated: 2026-09-26 19:48:21 UTC

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

(5085853, 1369757198)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-34, q=47; A=(3q-p)(p+q)^3=384475, B=-(p+3q)(p-q)^3=56864187. 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 19:48:21 UTC · history