  In this paper, we compute a formula for the $a$-number of curve $\mathbb{X}$
given by the equation $y^{q^2 + q + 1} = x^{q^2 + 1} - x^q$ over the finite
field $\mathbb{F}_{q^2}$. The $a$-number is an invariant of the isomorphism
class of the $p$-torsion group scheme. In this paper, we compute a closed
formula for the $a$-number of $\mathbb{X}$ using the action of the Cartier
operator on $H^0(\mathbb{X}, \Omega^1)$.
