The $a$-number is an invariant of the isomorphism class of the $p$-torsion group scheme. We use the Cartier operator on $H^0(\mathcal{A}_2,\Omega^1)$ to find a closed formula for the $a$-number of the form $\mathcal{A}_2 = v(Y^{\sqrt{q}}+Y-x^{\frac{\sqrt{q}+1}{2}})$ where $q=p^s$ over the finite field $\mathbb{F}_{q^2}$. The application of the computed $a$-number in coding theory is illustrated by the relationship between the algebraic properties of the curve and the parameters of codes that are supported by it.

翻译：$a$-数是对应于$p$-挠群概形同构类的不变量。我们利用$H^0(\mathcal{A}_2,\Omega^1)$上的卡蒂埃算子，给出了形如$\mathcal{A}_2 = v(Y^{\sqrt{q}}+Y-x^{\frac{\sqrt{q}+1}{2}})$（其中$q=p^s$）在有限域$\mathbb{F}_{q^2}$上的$a$-数的闭式公式。通过该曲线的代数性质与其所支持的码参数之间的关系，阐明了所计算的$a$-数在编码理论中的应用。