Automorphism group and twists of a pseudo-Fermat curve. I

For an integer $d\geq5$, let $\mathcal{C}/\mathbb{Z}$ be the scheme $x^d+y^d+xz^{d-1}=0$. We first show that the fibres $C_{d,0}:=\mathcal{C}\otimes\overline{\mathbb{Q}}$ and $C_{d,p}:=\mathcal{C}\otimes\overline{\mathbb{F}}_p$ for $p\nmid d(d-1)$ are smooth plane curves having the same automorphism group. We then provide the description of the twists of $\mathcal{C}\otimes\mathbb{R}$ over $\mathbb{R}$, and these of $\mathcal{C}\otimes\mathbb{F}_p$ over $\mathbb{F}_p$. In doing so, it becomes evident that for all the twists $C'$ of $\mathcal{C}\otimes\mathbb{Q}$, $C'(\mathbb{Q})\neq\emptyset$, as long as $d$ is not divisible by $3$.