curve #2198

y² + xy = x³ − 103909674963276x + 406864671310484084880

a-invariants: [1, 0, 0, -103909674963276, 406864671310484084880]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 1313542230
discriminant (Δ): 291092496811796384029191730644196416000000
Faltings height: 6.8364
naive height: 108.4372
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 31, 83
regulator: 4.011023050457090261078550409177408995752
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:46:32 UTC
last updated: 2026-09-26 19:46:32 UTC

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

(6102486, 123318000)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-31, q=52; A=(3q-p)(p+q)^3=1731807, B=-(p+3q)(p-q)^3=71473375. 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 19:46:33 UTC · history