On the Automorphism Group of a Binary Form Associated with Algebraic Trigonometric Quantities

Let $F(x,y)$ be a binary form of degree at least three and non-zero discriminant. In this article we compute the automorphism group $\mathrm{Aut}F$ for four families of binary forms. The first two families that we are interested in are homogenizations of minimal polynomials of $2\cos \left(\frac{2\pi }{n}\right)$ and $2\sin \left(\frac{2\pi }{n}\right)$, which we denote by ${\mathrm{\Psi }}_n(x,y)$ and ${\mathrm{\Pi }}_n(x,y)$, respectively. The remaining two forms that we consider are homogenizations of Chebyshev polynomials of first and second kinds, denoted $T_n(x,y)$ and $U_n(x,y)$, respectively.