Relatives of the Hermitian curve

We introduce the notion of a relative of the Hermitian curve of degree $\sqrt{q}+1$ over ${\mathbb{F}}_q$, which is a plane curve defined by $$(x^{\sqrt{q}},y^{\sqrt{q}},z^{\sqrt{q}})A{}^t\text{\hspace{0.17em}⁣}(x,y,z)=0$$ with $A\in GL(3,{\mathbb{F}}_q)$, and study their basic properties, one of which is that the number of ${\mathbb{F}}_q$-points of any relative of the Hermitian curve of degree $\sqrt{q}+1$ is congruent to $1$ modulo $\sqrt{q}$. In the latter part of this paper, we classify those curves having two or more rational inflexions.