From set of numbers from $\Bbb S=\{0,1,\dots,m\}$, how many distinct $3\times 3$ unimodular matrices parametrized by $(a,b,c,d,e,f)\in\Bbb S^6$ of following type can one form? \begin{bmatrix} a^2 &ab &b^2\\ c^2 &cd &d^2\\ e^2 &ef &f^2\\ \end{bmatrix}

Is it at least $3m^{2+\beta}$ for some $\beta>0$ when $m\gg0$?

From comment below determinant is $$(ad-bc)(af-be)(cd-ef).$$

So how many $3$ tuples of $2\times 2$ matrices of following type with determinant being simultaneously $\pm1$ with entries from $\Bbb S$? $$\begin{bmatrix} a &c\\ b &d \end{bmatrix}\quad \begin{bmatrix} c &e\\ d &f \end{bmatrix}\quad \begin{bmatrix} e &a\\ f &b \end{bmatrix}$$

An example matrix: \begin{bmatrix} 1 &1 &1\\ 9 &6 &4\\ 4 &2 &1\\ \end{bmatrix} has determinant $-1$.

**Update:**

As determined below by Kantelope and Neil Strickland, rough asymptotics seem to be at least $3m^2$. Could this be improved to $3m^{2+\beta}$ for some $\beta>0$ when $m\gg0$?