curve #2650

y² + xy + y = x³ − 22447443649x − 291336410627584

a-invariants: [1, 0, 1, -22447443649, -291336410627584]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 159232710
discriminant (Δ): 687237094185179431761666867896100
Faltings height: 4.9915
naive height: 83.1169
primes of bad reduction: 2, 3, 5, 7, 13, 17, 47, 73
regulator: 22.2049297395412339579585304696165352520393
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:15:30 UTC
last updated: 2026-09-26 20:15:30 UTC

Witness: 2 independent points log in to add more points →

(-120887, 25666478)
(-69416, 30569768)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-34, q=13; A=(3q-p)(p+q)^3=-676053, B=-(p+3q)(p-q)^3=519115. 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 20:15:31 UTC · history