Orbifold points on Prym-Teichmüller Curves in genus 3

Teichmüller curves are special curves in the moduli space M_g that
exhibit interesting arithmetic and geometric properties. We calculate
the number and types of orbifold points for a class of Teichmüller
curves inside M_3, thus completing their topological classification.
Crucial for the determination of the orbifold points is the analysis of
families of genus 3 cyclic covers of degree 4 and 6, branched over four
points of P^1. As a side product of our study, we provide an explicit
description of the Jacobians and the Prym-Torelli images of these two
families.