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

For every prime power $q>2$ and every even integer $n\ge4$, we construct an $\mathbb{F}_{q^{2n}}$-maximal curve of genus $(q^2-1)q^n/2$ 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\ge3