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?

Look at the Riemann-Hurwitz [RH] formula carefully. I recommend Plane Algebraic Curves by G. Fisher, or Complex Algebraic Curves by Kirwan. All kinds of interesting things can happen, and 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 local branching at each branch point to be 2: as in $z \to z^2$. The total degree will then be $6 = 2*3$. Does the Riemann surface exist? If so, its genus must be 3, by RH.

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?

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?