Automorphisms of Plane Curves defined from Chebychev polynomials

We investigate the automorphism groups of the algebraic curves $$\mathcal{C}_d : y^d = \varphi_d(x),$$ where $\varphi_d(x)$ denotes the Chebyshev polynomial of degree $d$, defined over a field $k$ with $p:=\operatorname{char}(k) \nmid 2d$. We determine the full automorphism group of $\mathcal{C}_d$ in all the cases considered in this paper, namely for $d=4$, and more generally when $2d = p^r+1$ or $4d = p^r+1$. For all other $d>4$, Expectation~\ref{3.19} predicts what the automorphism group should be. As an application, we show that certain maximal curves of the same genus are not isomorphic.