curve #2320

y² + xy = x³ − x² − 677499803955x + 137211692850677025

a-invariants: [1, -1, 0, -677499803955, 137211692850677025]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 99633690
discriminant (Δ): 11769228783335671855617866339557872900
Faltings height: 5.8152
naive height: 93.3386
primes of bad reduction: 2, 3, 5, 13, 31, 41, 67
regulator: 6.200571829452239252191382808732496109823
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:54:12 UTC
last updated: 2026-09-26 19:54:12 UTC

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

(-411092, 588638339)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-54, q=13; A=(3q-p)(p+q)^3=-6409653, B=-(p+3q)(p-q)^3=-4511445. 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 19:54:13 UTC · history