For an integer n>2, I calculate or estimate invariants associated with the minimal polynomial tn(x) of tan(π/n) and its homogenization tn(x,y). For tn(x), invariants include constant and leading coefficients, as well as the determinant. For the binary form tn(x,y), I estimate the area of the fundamental domain {(x, y) in R2 : | tn(x, y)|≤1} and calculate the automorphism group Aut tn (i.e., the group of 2x2 matrices A with rational entries such that tn(Av) = t_n(v) for all v=(x,y)T). I also solve the Thue equations tn(x,y)=±1 in the case when n>3 is prime by applying the famous result of Bilu, Hanrot and Voutier on Lucas sequences.