curve #2187

y² + xy + y = x³ − 267957396351013x + 1683474795848731400156

a-invariants: [1, 0, 1, -267957396351013, 1683474795848731400156]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 28145072670
discriminant (Δ): 7012044003652514964329057886794071289062500
Faltings height: 7.0865
naive height: 111.2792
primes of bad reduction: 2, 3, 5, 11, 13, 17, 31, 59, 211
regulator: 7.358372126849506114616492007390163092531
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:45:48 UTC
last updated: 2026-09-26 19:45:48 UTC

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

(10222650, 3535138987)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-34, q=59; A=(3q-p)(p+q)^3=3296875, B=-(p+3q)(p-q)^3=115023051. 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 19:45:48 UTC · history