curve #2670

y² + xy + y = x³ − 221385174469x + 40093199391408476

a-invariants: [1, 0, 1, -221385174469, 40093199391408476]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 69233010
discriminant (Δ): 3186965986392926320902188100
Faltings height: 4.9382
naive height: 89.9831
primes of bad reduction: 2, 3, 5, 7, 11, 17, 41, 43
regulator: 22.1122141281074136827431163256862502668465
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:16:56 UTC
last updated: 2026-09-26 20:16:56 UTC

Witness: 2 independent points log in to add more points →

(-364511, 269177828)
(77271, 153088492)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-22, q=21; A=(3q-p)(p+q)^3=-85, B=-(p+3q)(p-q)^3=3259787. 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 20:16:56 UTC · history