curve #2192

y² + xy + y = x³ − x² − 252525371242127x + 1544118321571418014679

a-invariants: [1, -1, 1, -252525371242127, 1544118321571418014679]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 2616270930
discriminant (Δ): 593488963998184413714867761433513984000000
Faltings height: 6.9883
naive height: 111.1012
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 23, 43
regulator: 2.500142081683455469821696572868383774055
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:46:09 UTC
last updated: 2026-09-26 19:46:09 UTC

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

(9033759, 310382698)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-39, q=56; A=(3q-p)(p+q)^3=1016991, B=-(p+3q)(p-q)^3=110601375. 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 19:46:09 UTC · history