curve #2397

y² + xy + y = x³ − 167940637833x + 26480988477824056

a-invariants: [1, 0, 1, -167940637833, 26480988477824056]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 133461510
discriminant (Δ): 205689061592413864953327720562500
Faltings height: 5.1649
naive height: 89.1542
primes of bad reduction: 2, 3, 5, 7, 13, 19, 31, 83
regulator: 2.538463755144404751451446377267070943759
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:59:03 UTC
last updated: 2026-09-26 19:59:03 UTC

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

(-300760, 223276167)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-26, q=19; A=(3q-p)(p+q)^3=-28469, B=-(p+3q)(p-q)^3=2824875. 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 19:59:03 UTC · history