I'm trying to prove an exercise problem from Szamuely's "Galois Groups and Fundamental Groups", which is a version of belyi's theorem.
The normal version without the ramification indices of points above 1 was already proven in the book. The part I'm concerned about is constructing such a morphism $X \to P^1$ (thats étale everywhere except 0, 1, $\infty$) where points above 1 have ramification index 2 when assuming that X can be defined over $\overline{Q}$.
My idea was to use the version of beyli's theorem that was already proven to take a morphism $X \to P^1$ thats étale everywhere but at 0, 1 and $\infty$ and compose with a belyi function $P^1 \to P^1$ of the form $x^A(x-1)^B$ for some nonzero integers A and B, in the hope of decreasing the ramification index of one point above 1. If I can do this I am done, since the morphism $X \to P^1$ is finite, so this procedure can be repeated until the ramification indices are equal to 2.
However I do not know, how I would have to choose A and B to decrease the ramification index of some point above 1, if thats even possible. How would I go about doing that, or is there a different way to solve the exercise?