I have something quick that I'd like to clarify regarding the definition of the form class group.
A binary quadratic form is a polynomial $q(x,y) = ax^2 + bxy + cy^2$ in two variables $x$ and $y$, with coefficients $a,b$ and $c$ and discriminant $\Delta = b^2-4ac$. When $a,b,c \in \mathbb{Z}$, $q(x,y)$ is known as an integral binary quadratic form (often simply called a binary quadratic form, which is the convention we will use).
Which one of the following is the most accurate definition of the form class group $h(\Delta)$ ?
1) The set of equivalence classes of binary quadratic forms with a fixed discriminant $\Delta$.
2) The set of equivalence classes of positive definite binary quadratic forms with a fixed discriminant $\Delta$.
3) The set of equivalence classes of primitive , positive definite binary quadratic forms with a fixed discriminant $\Delta$.
In the various sources that I've studied, I've come across all three definitions, and I'm not sure which is the correct one.
Also, is it correct that, even in the most general case where we consider the set of equivalence classes of binary quadratic forms with a fixed discriminant $\Delta$, each equivalence class has a canonical representative given by a reduced binary quadratic form ? Finding this reduced binary quadratic form is what Gauss' reduction algorithm accomplishes -- but I'm not sure if you can do this at the most general level that I described, or if you need your binary quadratic forms that you work with to be positive definite, primitive, or both for this to work.
Thank you !