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Prym–Teichmüller curves   $W_{D}(4)$   constitute the main examples of known primitive Teichmüller curves in the moduli space   $\mathcal{M}_3$  . We determine, for each non-square discriminant   $D&gt;1$  , the number and type of orbifold points in   $W_{D}(4)$  . These results, together with the formulas of Lanneau–Nguyen and Möller for the number of cusps and the Euler characteristic, complete the topological characterisation of Prym–Teichmüller curves in genus 3. 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   $\mathbb{P}^1$  . As a side product of our study, we provide an explicit description of the Jacobians and the Prym–Torelli images of these two families, together with a description of the corresponding flat surfaces.

Orbifold Points on Prym–Teichmüller Curves in Genus 3