Amirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-2449Articles in Press20211010The $a$-number of maximal curves of third largest genus455310.22060/ajmc.2021.20511.1069ENVahidNouroziFaculty of Mathematics and Computer Science‎, ‎Amirkabir University of TechnologySaeidTafazolianDepartment of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic)Journal Article20210907The $a$-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 $y^q + y = x^{frac{q+1}{3}}$ and $sum_{t=1}^{s} y^{q/3^t}= x^{q+1}$ with $q = 3^s$ over the finite field $mathbb{F}_{q^2}$ using the action of the Cartier operator on $H^0(mathcal{C},Omega^1)$.