Given a map $f:\mathbb{CP^1} \leftarrow \mathbb{CP^1}$ by $f(z)=\frac{4z^2(z-1)^2}{(2z-1)^2}$ Find all branching points and their degrees.
If my calculation is correct I got 4 branching points and in each point degree of $f$ is 2. Also I have concluded that infinity is not a branching point. From Riemann Hurwitz formula I get that total branching number is 6 where degree of $f$ is 4. But if I directly compute total branching number I get 4. Where did I make a mistake?
$f(z_0) = \infty$ is in fact a branch point. There will be six ramification points $z_0$: $$\begin{matrix} z_0 & f(z_0) \\ \hline 0, 1 & 0 \\ \frac {1 \pm i} 2 & -1 \\ \frac 1 2, \infty & \infty \end{matrix},$$ each of index $2$. To see why this is the case, expand $f(z)$ around each $z_0$; the degree of the first non-constant term will be $\pm 2$.