curve #2493

y² + xy = x³ − x² − 41707685349x + 3278488193729793

a-invariants: [1, -1, 0, -41707685349, 3278488193729793]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 2709630
discriminant (Δ): 74537164200916031985562500
Faltings height: 4.5449
naive height: 84.9755
primes of bad reduction: 2, 3, 5, 7, 11, 17, 23
regulator: 4.433042596419160114810473393968768872189
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:05:02 UTC
last updated: 2026-09-26 20:05:02 UTC

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

(-175128, 72278319)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-18, q=17; A=(3q-p)(p+q)^3=-69, B=-(p+3q)(p-q)^3=1414875. 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 20:05:02 UTC · history