Automorphisms of Plane Curves defined from Chebychev polynomials

We investigate the automorphism groups of the algebraic curves $${\mathcal{C}}_d:y^d={\phi }_d(x),$$ where ${\phi }_d(x)$ denotes the Chebyshev polynomial of degree $d$, defined over a field $k$ with $p:=\mathrm{char}(k)\nmid 2d$. 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.