Binary Quadratic Forms

A binary quadratic form is a polynomial

$$q(x,y) = ax^2 + bxy + cy^2$$

with $a,b,c\in\mathbb{Z}$. For example $q(x,y) = x^2 + y^2$ is a binary quadratic form, and there is a simple criterion for whether or not an integer $n$ is of the form $n = q(x,y)$ for $x,y\in\mathbb{Z}$.

The discriminant $b^2 - 4ac$ of $q$ is congruent to either 0 or $1$ modulo $4$. Suppose $D$ is a negative discrimant and consider the set of equivalence classes of binary quadratic forms of discriminant $D$, where two forms $q_1$ and $q_2$ are equivalent if and only if there exists $g\in\SL_2(\mathbb{Z})$ such that $q_{\vert g} = r,$ where

$$q_{\vert g}(x,y) = q\left(g\left( \begin{matrix}x\\ y \end{matrix}\right)\right).$$

A reduced binary quadratic form is one for which $\vert b\vert\leq a\leq c$ and, in addition, when one of the two inequalities is an equality then $b\geq 0$. Every form is equivalent to exactly one reduced form, so it is possible to decide whether or not two forms are equivalent. Also, there are only finitely many equivalence classes of fixed discriminant $D<0$. This finite set has a natural group structure.