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For a negative discriminant of a binary quadratic form $D\equiv 0$ or $1(\mathrm{mod}4)$, the class number of a negative discriminant $h(D)$ counts proper equivalence classes of primitive binary quadratic forms that are positive definite and have discriminant $D$.

Uniqueness of the reduced positive definite binary quadratic form in each class makes this the number of reduced primitive forms. Enumeration of reduced binary quadratic forms proves finiteness and gives an explicit procedure. For example, $h(-4)=1$, represented by $[1,0,1]$, while $h(-39)=4$, with the four forms listed under reduced positive definite binary quadratic forms of discriminant minus thirty-nine.

Imprimitive forms do not enter $h(D)$. At $D=-32$, the reduced form $[2,0,4]$ has content two, so the three reduced forms listed under reduced forms of discriminant minus thirty-two contribute only two primitive classes. If $D$ is a fundamental discriminant, this form class number agrees with the class number of the imaginary quadratic field of discriminant $D$. For nonfundamental $D$, the associated order in a number field need not be the full ring of integers, so that distinction must be retained.

The discriminant of a binary quadratic form is $d=b^2-4ac$. If the form is positive-definite, evaluating it at $(1,0)$ gives $a>0$. For $a\ne 0$, completing the square givesEvaluating along $x=-by/(2a)$ shows that positivity also requires $d<0$. Conversely these two inequalities make both displayed coefficients positive, provingA reduced positive definite binary quadratic form satisfies $|b|\le a\le c$, with $b\ge 0$ if $|b|=a$ or $a=c$. The last convention avoids counting both boundary representatives. For fixed negative discriminant, these inequalities implyThere are finitely many such integers $a$, finitely many integers $b$ with $|b|\le a$, and then $c=(b^2-d)/(4a)$ is determined. Hence the number of reduced forms is finite. If no admissible integer coefficients have the given discriminant, this number is zero.

For $d=-39$, the bound gives $a=1,2,3$. The integrality of $c=(b^2+39)/(4a)$ and the boundary sign convention give respectivelyThese are all the reduced positive definite binary quadratic forms of discriminant minus thirty-nine. Thusrepresented by $x^2+xy+10y^2$, $2x^2+xy+5y^2$, $2x^2-xy+5y^2$, and $3x^2+3xy+4y^2$.