Integral binary quadratic form

An integral binary quadratic form is a polynomial

f(x,y)=ax^2+bxy+cy^2,\qquad a,b,c\in\mathbb Z.

Its discriminant is $D=b^2-4ac$ . The form is primitive if $\gcd(a,b,c)=1$ , and positive definite if $f(x,y)>0$ for every nonzero $(x,y)\in\mathbb R^2$ , equivalently $a>0$ and $D<0$ . It is a quadratic form on $\mathbb Q^2$ together with its integral coordinate lattice.

Normalization

The middle coefficient is $b$ , not $2b$ . The symmetric matrix of the form is $\begin{pmatrix}a&b/2\\b/2&c\end{pmatrix}$ , so its determinant is $-D/4$ . Confusing these two middle-coefficient conventions changes the discriminant.

For instance, $x^2+xy+y^2$ is primitive positive definite of discriminant $-3$ . Integer changes of variables with determinant $1$ give proper equivalence.

References

Andrew V. Sutherland, 18.785 Number Theory, Fall 2018, Problem Set 7, Problem 4, initial definitions.