Orbifold points on Prym-Teichm\"uller curves in genus three

Prym-Teichm\"uller curves $W_D(4)$ constitute the main examples of known primitive Teichm\"uller curves in the moduli space $\mathcal{M}_3$. We determine, for each non-square discriminant $D>1$, the number and type of orbifold points in $W_D(4)$. These results, together with the formulas of Lanneau-Nguyen and M\"oller for the number of cusps and the Euler characteristic, complete the topological characterisation of Prym-Teichm\"uller 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.