1.I.8B

Let $q(x, y)=a x^{2}+b x y+c y^{2}$ be a binary quadratic form with integer coefficients. Define what is meant by the discriminant $d$ of $q$, and show that $q$ is positive-definite if and only if $a>0>d$. Define what it means for the form $q$ to be reduced. For any integer $d<0$, we define the class number $h(d)$ to be the number of positive-definite reduced binary quadratic forms (with integer coefficients) with discriminant $d$. Show that $h(d)$ is always finite (for negative $d)$. Find $h(-39)$, and exhibit the corresponding reduced forms.

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