curve #2311

y² + xy = x³ − 3051334087500x + 2051197041440250000

a-invariants: [1, 0, 0, -3051334087500, 2051197041440250000]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 65934330
discriminant (Δ): 630559943463009495824744841216000000
Faltings height: 5.8676
naive height: 97.8534
primes of bad reduction: 2, 3, 5, 7, 11, 17, 23, 73
regulator: 3.071875293987179679512850274990029797904
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:53:41 UTC
last updated: 2026-09-26 19:53:41 UTC

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

(973080, 57820020)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-23, q=32; A=(3q-p)(p+q)^3=86751, B=-(p+3q)(p-q)^3=12145375. 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 19:53:41 UTC · history