curve #2226

y² + xy = x³ − 21800691588666x − 15759776661394326300

a-invariants: [1, 0, 0, -21800691588666, -15759776661394326300]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 6140896230
discriminant (Δ): 555821897393891176177063635631157339750400
Faltings height: 6.7036
naive height: 103.7525
primes of bad reduction: 2, 3, 5, 7, 17, 61, 163, 173
regulator: 8.380818812153501175905179111909679842620
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:48:17 UTC
last updated: 2026-09-26 19:48:17 UTC

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

(-3026781/4, 4409776947/8)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-5, q=56; A=(3q-p)(p+q)^3=22948623, B=-(p+3q)(p-q)^3=36997903. 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 19:48:18 UTC · history