I've been back on researching the task again. I thought I might share some new results, if they are not known already:
x^5 + (n^2 + n + 4)*x^4 + (2*n^2  2*n + 2)*x^3 + (n^4  2*n^3  10*n^2  9*n  5)*x^2 + (3*n^2  3*n  2)*x + n^2 + n + 1
x^7 + (n^3  3*n^2  4*n  1)*x^6 + (3*n^5 + 9*n^3 + 3*n^2 + 3*n  12)*x^5 + (3*n^7 + 6*n^6  4*n^5 + n^4  7*n^3 + 28*n^2 + 15*n + 7)*x^4 + (n^9  5*n^8  2*n^7  5*n^6 + 6*n^5  11*n^4  21*n^3 + 12*n^2  n + 28)*x^3 + (n^10 + n^9 + 2*n^8  n^7  9*n^6 + 4*n^5  32*n^4  6*n^3  49*n^2 + 5*n  14)*x^2 + (n^9 + 3*n^8  n^7 + 11*n^6 + 4*n^5 + 4*n^4  10*n^3  30*n^2  20*n  9)*x  3*n^7 + 3*n^6  7*n^5 + 14*n^4  5*n^3 + 12*n^2  6*n  1
As far as I know, there is only one other known cyclic septic besides the one I have listed here.
