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?
Richard Montgomery
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