Some remarks about the minimal polynomials of $4{\sin }^2\left(\frac{\pi }{n}\right)$

We show that the values of the minimal polynomials ${\phi }_n(x)$ of $4{\sin }^2\left(\frac{\pi }{n}\right)$ for $x\in \left\{0,1,2,3,4\right\}$ are intimately related to the prime factorization of $n\mathrm{.}$