curve #2199

y² + xy + y = x³ − x² − 116867762420567x + 485792659633573720559

a-invariants: [1, -1, 1, -116867762420567, 485792659633573720559]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 1193061870
discriminant (Δ): 206450972407865730924482032247782464000000
Faltings height: 6.8396
naive height: 108.7898
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 41
regulator: 1.905135106379449815709589644954579555140
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:46:36 UTC
last updated: 2026-09-26 19:46:36 UTC

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

(6480267, 764712766)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-33, q=52; A=(3q-p)(p+q)^3=1296351, B=-(p+3q)(p-q)^3=75537375. 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 19:46:36 UTC · history