curve #2188

y² + xy = x³ − 422809218892040x + 3345020662233716289600

a-invariants: [1, 0, 0, -422809218892040, 3345020662233716289600]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 72508206030
discriminant (Δ): 3696061696198034282181940839262157376000000
Faltings height: 7.1267
naive height: 112.6474
primes of bad reduction: 2, 3, 5, 13, 17, 19, 41, 101, 139
regulator: 3.858166166920637946245239915664994678965
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:45:53 UTC
last updated: 2026-09-26 19:45:53 UTC

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

(12334072, 2530659256)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-41, q=60; A=(3q-p)(p+q)^3=1515839, B=-(p+3q)(p-q)^3=143211839. 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 19:45:53 UTC · history