curve #2207

y² + xy + y = x³ − x² − 63269375049302x + 192706053003846837029

a-invariants: [1, -1, 1, -63269375049302, 192706053003846837029]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 385564410
discriminant (Δ): 166577245771067595151708808283890625000000
Faltings height: 6.7495
naive height: 106.9489
primes of bad reduction: 2, 3, 5, 7, 11, 23, 41, 59
regulator: 6.036298056353334215725661474809771854676
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:47:06 UTC
last updated: 2026-09-26 19:47:06 UTC

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

(19944523/4, 8741230973/8)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-27, q=50; A=(3q-p)(p+q)^3=2153559, B=-(p+3q)(p-q)^3=56153559. 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 19:47:06 UTC · history