Division and Binary Quadratic Forms

This all must go back to people such as Legendre, Dirichlet, and Gauss. I have no idea, really. We use 〈α, β, γ 〉 to denote the (positive) binary quadratic form f(x, y) = αx2 + βxy + γy2. The discriminant ∆ is given by ∆ = β2 − 4αγ and so is negative for positive forms. We insist that our forms be primitive, that is gcd(α, β, γ) = 1. For discriminant ∆ = −23, the entire class grou