A new family of maximal curves not covered by the Hermitian curve

For every prime power $q>2$ and every even integer $n\ge 4$, we construct an ${\mathbb{F}}_{q^{2n}}$-maximal curve of genus $(q^2-1)q^n\mathrm{/}2$ that is not covered by the Hermitian curve over ${\mathbb{F}}_{q^{2n}}$. The defining equation also gives a Kummer model for the Beelen–Montanucci curves when $n\ge 3$ is odd. We compute the genus for both odd and even $n$ and give a uniform proof of maximality by counting rational places. For $q>2$ and odd $n\ge 5$, we also prove that the Beelen–Montanucci curves are not covered by the Hermitian curve, extending the known result for Galois coverings. For every prime power $q$ and $n\ge 4$, we also give an explicit automorphism subgroup of order $(q^n+1)q(q^2-1)$.