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.
$$x_1=2\left(\cos\frac{2\pi}{43}+\cos\frac{12\pi}{43}+\cos\frac{14\pi}{43}\right),$$
$$x_2=2\left(\cos\frac{6\pi}{43}+\cos\frac{36\pi}{43}+\cos\frac{42\pi}{43}\right),$$
$$x_3=2\left(\cos\frac{18\pi}{43}+\cos\frac{22\pi}{43}+\cos\frac{40\pi}{43}\right)$$
$$x_4=2\left(\cos\frac{20\pi}{43}+\cos\frac{32\pi}{43}+\cos\frac{34\pi}{43}\right),$$
$$x_5=2\left(\cos\frac{10\pi}{43}+\cos\frac{16\pi}{43}+\cos\frac{26\pi}{43}\right),$$
$$x_6=2\left(\cos\frac{8\pi}{43}+\cos\frac{30\pi}{43}+\cos\frac{38\pi}{43}\right)$$ and
$$x_7=2\left(\cos\frac{4\pi}{43}+\cos\frac{24\pi}{43}+\cos\frac{28\pi}{43}\right).$$
I see only that $\sum\limits_{k=1}^7x_k=-1$, but the computations for $\sum\limits_{1\leq i<j\leq7}x_ix_j$ 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!
Update.
I got $$\sum\limits_{1\leq i<j\leq7}x_ix_j=-18.$$

By PARI / GP I get
$x^7 + x^6 - 18*x^5 - 35*x^4 + 38*x^3 + 104*x^2 + 7*x - 49$ :
K = nfinit (subst(polcyclo(43),x,y))
w = Mod(y,K.pol)
f0(k) = (w^k + 1/w^k)
f(k1,k2,k3) = f0(k1) + f0(k2) + f0(k3)
v = [f(1,6,7),f(3,18,21),f(9,11,20),f(10,16,17),f(5,8,13),f(4,15,19),f(2,12,14)]
/*
=
[x^7 + x^6 - 18x^5 - 35x^4 + 38x^3 + 104x^2 + 7*x - 49,
x^7 + x^6 - 18x^5 - 35x^4 + 38x^3 + 104x^2 + 7*x - 49,
x^7 + x^6 - 18x^5 - 35x^4 + 38x^3 + 104x^2 + 7*x - 49,
x^7 + x^6 - 18x^5 - 35x^4 + 38x^3 + 104x^2 + 7*x - 49,
x^7 + x^6 - 18x^5 - 35x^4 + 38x^3 + 104x^2 + 7*x - 49,
x^7 + x^6 - 18x^5 - 35x^4 + 38x^3 + 104x^2 + 7*x - 49,
x^7 + x^6 - 18x^5 - 35x^4 + 38x^3 + 104x^2 + 7*x - 49]
*/
mps = [minpoly(w) | w<-v]
