It was brought up in this question (Distribution of 'square classes' of binary quadratic forms) that the objects I am interested in are actually binary quadratic forms in the principal genus of its discriminant.
One can then formulate he following question. Consider the set of integral points
$$\displaystyle \{(a,b,c) \in \mathbb{Z}^3 : \max\{|a|, |b|, |c|\} \leq X\}$$
and the map $\rho$ from $\mathbb{Z}^3$ to the lattice of integral binary quadratic forms sending $(a,b,c)$ to $ax^2 + bxy + cy^2$.
We say an integral binary quadratic form is in the principal genus of discriminant $d$ if it is $\text{GL}_2(\mathbb{R})$-equivalent to the principal form of discriminant $d$ and it is $\text{GL}_2(\mathbb{Z}_p)$-equivalent to the principal form for every prime $p$.
What is the asymptotic size of the following two sets?
$$\displaystyle S(X) = \{(a,b,c) \in \mathbb{Z}^3 : \rho(a,b,c) \text{ is in the principal genus}, \max\{|a|, |b|, |c|\} \leq X\}$$
and
$$\displaystyle S_a(X) = \{(b,c) \in \mathbb{Z}^2: \rho(a,b,c) \text{ is in the principal genus}, \max\{|b|, |c|\} \leq X\}$$
