I have the curve $C \subset \mathbb{C}^2$ defined by the equation $y^3 + y^6 = x^6$ and I have to prove that the set of maps of the form $\varphi = (f_1,f_2) \in \operatorname{Aut}(\mathbb{C}^2)$ such that $\deg(f_1) = \deg(f_2) = 1$ and $\varphi (C) = C$ is a group isomophic to $\mathbb{Z}/6\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$.
It's clear that $\left\{ \varphi (x,y) = \left(e^{\frac{2\pi i}{6} k} x,e^{\frac{2\pi i}{3} k'} y \right): 1 \leqslant k \leqslant 6,1 \leqslant k' \leqslant 3\right\}$ holds all conditions and it's isomophic to $\mathbb{Z}/6\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$, but I can't prove that these morphisms are all.
Can somebody give me a guidance?