  For each discriminant $D>1$, McMullen constructed the Prym-Teichm\"uller
curves $W_D(4)$ and $W_D(6)$ in $\mathcal{M}_{3}$ and $\mathcal{M}_{4}$, which
constitute one of the few known infinite families of geometrically primitive
Teichm\"{u}ller curves. In the present paper, we determine for each $D$ the
number and type of orbifold points on $W_D(6)$. These results, together with a
previous result of the two authors in the genus $3$ case and with results of
Lanneau-Nguyen and M\"oller, complete the topological characterisation of all
Prym-Teichm\"uller curves and determine their genus.
  The study of orbifold points relies on the analysis of intersections of
$W_D(6)$ with certain families of genus $4$ curves with extra automorphisms. As
a side product of this study, we give an explicit construction of such families
and describe their Prym-Torelli images, which turn out to be isomorphic to
certain products of elliptic curves. We also give a geometric description of
the flat surfaces associated to these families and describe the asymptotics of
the genus of $W_D(6)$ for large $D$.
