curve #2196

y² + xy + y = x³ − 355409891958854x + 2578832916073844518856

a-invariants: [1, 0, 1, -355409891958854, 2578832916073844518856]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 44957737110
discriminant (Δ): 253645525413167139293578878217079103872400
Faltings height: 7.0145
naive height: 112.1265
primes of bad reduction: 2, 3, 5, 11, 13, 19, 43, 101, 127
regulator: 5.949778603615869161059353908776824122041
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:46:25 UTC
last updated: 2026-09-26 19:46:25 UTC

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

(10816349, 187004097)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-44, q=57; A=(3q-p)(p+q)^3=472355, B=-(p+3q)(p-q)^3=130848227. 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 19:46:25 UTC · history