The Class Number H(d) of a Discriminant d

# The Class Number H(d) of a Discriminant d

Recall from the Equivalent Binary Quadratic Forms page that two binary quadratic forms $f(x, y) = ax^2 + bxy + cy^2$ and $g(x, y) = Ax^2 + Bxy + Cy^2$ are said to be equivalent denoted $f \sim g$ if there exists integers $m_{11}, m_{12}, m_{21}, m_{22} \in \mathbb{Z}$ such that:

(1)
\begin{align} \quad f(x, y) = g(m_{11}x + m_{12}y, m_{21}x + m_{22}y) \end{align}

and $m_{11}m_{22} - m_{12}m_{22} = 1$. We saw that $\sim$ is an equivalence relation on the set of binary quadratic forms. Furthermore, we noted that if $f \sim g$ then $f$ and $g$ have the same discriminant. However, recall that $f$ and $g$ having the same discriminant does not imply that $f \sim g$.

Given any discriminant $d$ there may be many equivalence classes which correspond to $d$. We define the number of such equivalence classes below.

 Definition: The Class Number of the discriminant $d$ denoted $H(d)$ (or $h(d)$) is the number of equivalence classes for the equivalence relation $\sim$ on the set of binary quadratic forms whose representative for the equivalence class has discriminant $d$.

If $d \equiv 0, 1 \pmod 4$ and $d < 0$ then it can be shown that the class number $H(d)$ is the number of nonequivalent binary quadratic forms with discriminant $d$. The first few class numbers are given in the table below.

 $d$ $H(d)$ $-3$ $-4$ $-7$ $-8$ $-11$ $-12$ $-15$ $-16$ $-19$ $-20$ $1$ $1$ $1$ $1$ $1$ $2$ $2$ $2$ $1$ $2$