curve #2222

y² + xy + y = x³ − x² − 25401661821608x + 48766688115388837931

a-invariants: [1, -1, 1, -25401661821608, 48766688115388837931]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 3224706030
discriminant (Δ): 21600204349176429668675888390614753934400
Faltings height: 6.5509
naive height: 104.2111
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 23, 53
regulator: 3.339036684360689327208539158167504488053
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:48:03 UTC
last updated: 2026-09-26 19:48:03 UTC

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

(2633625, 366111157)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-57, q=34; A=(3q-p)(p+q)^3=-1934553, B=-(p+3q)(p-q)^3=33910695. 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 19:48:03 UTC · history