curve #2191

y² + xy = x³ − x² − 194120678623464x + 1040295595732515182448

a-invariants: [1, -1, 0, -194120678623464, 1040295595732515182448]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 4931436510
discriminant (Δ): 644351188018896289853475274062542450250000
Faltings height: 6.9526
naive height: 110.3121
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 43, 67
regulator: 4.277297760151124856246391460962681351932
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:46:05 UTC
last updated: 2026-09-26 19:46:05 UTC

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

(8253012, 584190444)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-36, q=55; A=(3q-p)(p+q)^3=1378659, B=-(p+3q)(p-q)^3=97210659. 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 19:46:05 UTC · history