curve #2229

y² + xy = x³ − 89629421863290x + 326591007583901552100

a-invariants: [1, 0, 0, -89629421863290, 326591007583901552100]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 4020008190
discriminant (Δ): 4196094329748306610904822448390144000000
Faltings height: 6.6711
naive height: 107.9937
primes of bad reduction: 2, 3, 5, 11, 17, 37, 107, 181
regulator: 2.312304455785677762136226839585883378143
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:48:28 UTC
last updated: 2026-09-26 19:48:28 UTC

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

(5428470, 87267360)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-37, q=48; A=(3q-p)(p+q)^3=240911, B=-(p+3q)(p-q)^3=65711375. 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 19:48:28 UTC · history