curve #2223

y² + xy + y = x³ − 23288324921758x + 38818780940691591056

a-invariants: [1, 0, 1, -23288324921758, 38818780940691591056]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 641230590
discriminant (Δ): 157361332606169725304029078170606756000000
Faltings height: 6.6377
naive height: 103.9505
primes of bad reduction: 2, 3, 5, 7, 11, 13, 131, 163
regulator: 3.240175721120154869198050188258297218308
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:48:06 UTC
last updated: 2026-09-26 19:48:06 UTC

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

(3810565, 2323617917)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-16, q=49; A=(3q-p)(p+q)^3=5857731, B=-(p+3q)(p-q)^3=35975875. 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 19:48:07 UTC · history