curve #2438

y² + xy + y = x³ − 16610989608x + 649289808608806

a-invariants: [1, 0, 1, -16610989608, 649289808608806]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 82492410
discriminant (Δ): 111214610938396503947579420250000
Faltings height: 4.8627
naive height: 82.2136
primes of bad reduction: 2, 3, 5, 7, 11, 13, 41, 67
regulator: 6.767923996229997032576763866113278517827
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:01:36 UTC
last updated: 2026-09-26 20:01:36 UTC

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

(116945, 17436567)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-28, q=13; A=(3q-p)(p+q)^3=-226125, B=-(p+3q)(p-q)^3=758131. 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 20:01:37 UTC · history