curve #2306

y² + xy = x³ − 1164940169721x + 338648105509531065

a-invariants: [1, 0, 0, -1164940169721, 338648105509531065]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 64127910
discriminant (Δ): 51636220385698395361387621800205646400
Faltings height: 5.9422
naive height: 94.9647
primes of bad reduction: 2, 3, 5, 7, 11, 17, 23, 71
regulator: 2.962269791837816652702122570343690638614
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:53:23 UTC
last updated: 2026-09-26 19:53:23 UTC

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

(-1173912, 297998301)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-49, q=22; A=(3q-p)(p+q)^3=-2263545, B=-(p+3q)(p-q)^3=6084487. 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 19:53:23 UTC · history