Let $X$ be Klein's quartic curve given by $x^3y + y^3z+z^3x=0$ in $\mathbf{P}^2$. It is isomorphic to $X(7)$.
How do I easily show that $X$ is not hyperelliptic?
I can see that $X$ is of genus $3$ and has gonality $\leq 3$ (consider the projection). I'm trying to prove that it has gonality $3$.
More generally, what is a computationally feasible way to check if a curve is not hyperelliptic?
Note that I'm not really asking for a criterion. For example, to check if a variety is normal you could try to show that it is regular (which is easier to me).
Is the obvious morphism $X\to \mathbf{P}^1$ of degree $3$ Galois? That is, do we have that $X$ is a cyclic cover of degree $3$?