In the following problem
Let $M=\bigg (\begin{matrix} a& b\\ c& d\end{matrix}\bigg )$ be a nonsingular matrix with integer coefficients and $L=\mathbb C(x,y)$.
(i) Show that $\phi(x)=x^a y^b$, $\phi(y)=x^cy^d$ can be uniquely extended to a field homomorphism from $L$ to $L$, with $\phi|_\mathbb C=id$.
(ii) Show that $[L:\phi(L)]$ (algebraic degree) is finite and that $\phi$ is an automorphism if and only if $\det M=\pm 1$.
(iii) If $\phi$ is an automorphism, show that $\phi$ has non-constant fixed points if and only if $M$ has an eigenvalue which is a root of unity.
(iv) In the situation of (iii), show that either $\phi$ is conjugate to $\pm\bigg(\begin{matrix}1 & 1\\ 0 & 1\end{matrix}\bigg )$ or $\phi^{12}=id$.
(v) Show that, if $K$ is the set of the points fixed by $\phi$, then $K$ is a field and $\mathrm{trdeg}(K/\mathbb C)$ is $1$ or $2$ ($1$ in the first case of (iv), and $2$ in the second case).
I have some problems in understanding "$\phi^{12}=id$". I solved the whole problem and demonstrated, instead of that, that if $\phi$ is not conjugate to the given Jordan block, then it has diagonal form and its eigenvalues are roots of $1$ or $-1$ (of the same order), and therefore that $\phi^n=id$ for some $n$: but why 12?
However, my statement of point (iv) allows me to prove point (v) too.