On some Galois covers of the Suzuki and Ree curves

Abstract: We determine the full automorphism group of two recently constructed families $\tilde{\mathcal{S}}q$ and $\tilde{\mathcal{R}}_q$ of maximal curves over finite fields. These curves are covers of the Suzuki and Ree curves, and are analogous to the Giulietti-Korchm\'aros cover of the Hermitian curve. We also show that $\tilde{\mathcal{S}}_q$ is not Galois covered by the Hermitian curve maximal over $\mathbb{F}{q^4}$, and $\tilde{\mathcal{R}}q$ is not Galois covered by the Hermitian curve maximal over $\mathbb{F}{q^6}$. Finally, we compute the genera of many Galois subcovers of $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$; this provides new genera for maximal curves.