By "complex structure" I am referring to 2x2 matrices which square to $-\mathrm{Id}_2$. I need to know those with integer entries and determinant equal to 1.
Thank you
What are all the complex structures on $\mathbb{R}^2$ which live inside $\mathrm{SL}_2(\mathbb{Z})$?
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The matrices in $\mathrm{SL}_2(\mathbb{Z})$ whose square is $-\mathrm{Id}_2$ are precisely the matrices in $\mathrm{SL}_2(\mathbb{Z})$ with characteristic polynomial $x^2+1$. So these are the matrices in $\mathrm{SL}_2(\mathbb{Z})$ whose trace is zero. By looking at the corresponding fixed points in the upper half-plane, it is not hard to show that the matrices in question form two conjugacy classes in $\mathrm{SL}_2(\mathbb{Z})$ with representative elements $$\begin{pmatrix}0&-1\\1&0\end{pmatrix}\qquad\text{and}\qquad\begin{pmatrix}0&1\\-1&0\end{pmatrix}.$$