On a Galois cover of the Hermitian curve of genus \(\mathfrak{g}=\frac{1}{8}(q-1)^2\)

In the study of algebraic curves with many points over a finite field, a well known general problem is to understanding better the properties of \(\mathbb{F}_{q^2}\)-maximal curves whose genera fall in the higher part of the spectrum of the genera of all \(\mathbb{F}_{q^2}\)-maximal curves. This problem is still open for genera smaller than \( \lfloor \frac{1}{6}(q^2-q+4) \rfloor\). In this paper we consider the case of \(\mathfrak{g}=\frac{1}{8}(q-1)^2\) where \(q\equiv 1\pmod{4}\) and the curve is the Galois cover of the Hermitian curve w.r.t to a cyclic automorphism group of order \(4\). Our contributions concern Frobenius embedding, Weierstrass semigroups and automorphism groups.