You ought to look at the Riemann-Hurwitz formula carefully.
I recommend _Plane Algebraic Curves_ by G. Fisher, or _Complex Algebraic Curves_
by Kirwan. All kinds of interesting things can happen, in symmetric ways,
if 4 divides n. For example, you might try to have the inverse image of each
of branch point be exactly 3 points, and look for the branching degree at each branch point to be 2: as in $z \to z^2$.  The total degree will then be 6. 
Does the Riemann surface exist?