curve #2321

y² + xy + y = x³ − 659272817078x + 146338171788085256

a-invariants: [1, 0, 1, -659272817078, 146338171788085256]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 131741610
discriminant (Δ): 9087766203180685592656475015625000000
Faltings height: 5.7981
naive height: 93.2568
primes of bad reduction: 2, 3, 5, 7, 11, 13, 41, 107
regulator: 3.791401937410645200341138120193386500048
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:54:16 UTC
last updated: 2026-09-26 19:54:16 UTC

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

(-844995, 316776997)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-8, q=33; A=(3q-p)(p+q)^3=1671875, B=-(p+3q)(p-q)^3=6271811. 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 19:54:16 UTC · history