Minimal Polynomial Stuff

If $t = \tan\tfrac{\pi}{15}$ , then $\frac{t^5 - 10t^3 + 5t}{5t^4 - 10t^2 + 1} \; = \; \tan\big(5\tfrac{\pi}{15}\big) \; = \; \tan\tfrac13\pi \; = \; \tfrac{1}{\sqrt{3}}$ and hence $0 \; = \; 3(t^5 - 10t^3 + 5t)^2 - (5t^4 - 10t^2 + 1)^2 \; = \; (3t^2 - 1)(t^8 - 28t^6 + 134t^4 - 92t^2 + 1)$ which means that the minimum polynomial of $t$ certainly divides $X^8 - 28X^6 + 134X^4 - 92X^2 + 1$ . It turns out that this last polynomial is in fact irreducible, and hence is the minimum polynomial of $t$ . Thus the answer is $\boxed{134}$ .