# Similar reduced integral matrices

Let $$d\geq 1$$ be an integer. I'm not assuming anything here about $$d$$ (it is not necessarily squarefree, for example)

For all $$a,b,c\in\mathbb{Z}$$ such that $$ac-b^2=d,$$ set $$[a,b,c]_d=\begin{pmatrix}b & -c \cr a & -b\end{pmatrix}\in M_2(\mathbb{Z})$$.

(Note that the relation $$ac-b^2=d$$ forces $$a\neq 0$$ and $$c\neq 0$$.)

Let's say that $$[a,b,c]_d$$ is reduced if $$0, and $$b\geq 0$$ whenever $$a=c$$.

Note the similarity with the notion of reduced positive binary form à la Gauss (up to a factor $$1/2$$ which might be easily explained); this is not a coincidence.

Question. Without using the theory of binary forms, is it possible to prove that, if $$[a,b,c]_d$$ and $$[a',b,'c']_d$$ are reduced and similar over $$\mathbb{Z}$$ (that is, conjugate by an element of $$GL_2(\mathbb{Z})$$), then $$a=a',b=b',c=c'$$.

[Let me explain why it should be true. The set of conjugacy classes of integral matrices with characteristic polynomial $$X^2+d$$ and the set of equivalences of ideals of $$\mathbb{Z}[\sqrt{-d}]$$, where two ideals $$I$$ and $$J$$ are equivalent if there exist $$z,z'\in \mathbb{Q}(\sqrt{-d})$$ such that $$zI=z'J$$.

But if I am not mistaken, it is known that the set of classes of ideals is in bijection with the set of equivalent positive definite binary forms of discriminant $$-4d$$. And such a form is properly equivalent to a unique reduced binary form. Now, my matrix $$[a,b,c]_d$$ is reduced if and only if $$ax^2+2bxy+cy^2$$ is reduced.]