curve #2221

y² + xy = x³ − x² − 39176758818855x + 94271073998895938025

a-invariants: [1, -1, 0, -39176758818855, 94271073998895938025]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 1076665590
discriminant (Δ): 9071045513874506684513380942404552600900
Faltings height: 6.5721
naive height: 105.5109
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 37
regulator: 6.335558136784002707542214431925877895449
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:47:59 UTC
last updated: 2026-09-26 19:47:59 UTC

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

(-7221564, 763096509)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-54, q=37; A=(3q-p)(p+q)^3=-810645, B=-(p+3q)(p-q)^3=42953547. 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 19:47:59 UTC · history