curve #2390

y² + xy = x³ − 82249537811x − 8813752919926815

a-invariants: [1, 0, 0, -82249537811, -8813752919926815]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 84584010
discriminant (Δ): 2052012954057154341796000189569600
Faltings height: 5.1648
naive height: 87.0127
primes of bad reduction: 2, 3, 5, 7, 17, 19, 29, 43
regulator: 5.424744166050373990494066665202011624599
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:58:35 UTC
last updated: 2026-09-26 19:58:35 UTC

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

(-179816, 12814081)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-43, q=14; A=(3q-p)(p+q)^3=-2073065, B=-(p+3q)(p-q)^3=-185193. 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 19:58:35 UTC · history