curve #2239

y² + xy + y = x³ − 16226136984223x − 22103022687740160994

a-invariants: [1, 0, 1, -16226136984223, -22103022687740160994]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 9045810510
discriminant (Δ): 62366106360942503694120801918421557562500
Faltings height: 6.5558
naive height: 102.8665
primes of bad reduction: 2, 3, 5, 11, 19, 53, 163, 167
regulator: 7.503928483007609609556159187846323227725
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:49:05 UTC
last updated: 2026-09-26 19:49:05 UTC

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

(-2917823, 634334528)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-2, q=55; A=(3q-p)(p+q)^3=24862459, B=-(p+3q)(p-q)^3=30186459. 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 19:49:06 UTC · history