The $a$-number of $y^{q^2+q+1}=x^{q^2+1}+x^q$ over finite field

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 $H^0(\mathbb{X},{\mathrm{\Omega }}^1)$.