The Rank of the Cartier operator on Picard Curves

For an algebraic curve $\mathcal{X}$ defined over an algebraically closed field of characteristic $p>0$, the $a$-number $a(\mathcal{X})$ is the dimension of the space of exact holomorphic differentials on $\mathcal{X}$. We compute the $a$-number for a family of certain Picard curves, using the action of the Cartier operator on $H^0(\mathcal{X},{\mathrm{\Omega }}^1)$.