# galois theory – Minimal polynomial in \$mathbb Z[x]\$ of seventh degree with given roots

I am looking for a seventh degree polynomial with integer coefficients, which has the following roots.
$$x_1=2left(cosfrac{2pi}{43}+cosfrac{12pi}{43}+cosfrac{14pi}{43}right),$$
$$x_2=2left(cosfrac{6pi}{43}+cosfrac{36pi}{43}+cosfrac{42pi}{43}right),$$
$$x_3=2left(cosfrac{18pi}{43}+cosfrac{22pi}{43}+cosfrac{40pi}{43}right)$$
$$x_4=2left(cosfrac{20pi}{43}+cosfrac{32pi}{43}+cosfrac{34pi}{43}right),$$
$$x_1=2left(cosfrac{10pi}{43}+cosfrac{16pi}{43}+cosfrac{26pi}{43}right),$$
$$x_1=2left(cosfrac{8pi}{43}+cosfrac{30pi}{43}+cosfrac{38pi}{43}right)$$ and
$$x_1=2left(cosfrac{4pi}{43}+cosfrac{24pi}{43}+cosfrac{28pi}{43}right).$$
I see only that $$sumlimits_{k=1}^7x_k=-1$$, but the computations for $$sumlimits_{1leq i and the similar are very complicated by hand and I have no any software besides WA, which does not help.

Thank you for your help!