% orbit 6, S = (0,-1,1,0), degree 7
\begin{align*}
x &= 27m^{7} - 108m^{6}n + 189m^{5}n^{2} - 152m^{4}n^{3} + 47m^{3}n^{4} + 10m^{2}n^{5} - 11mn^{6} + 2n^{7},\\
y &= -9m^{7} - 3m^{6}n + 91m^{5}n^{2} - 169m^{4}n^{3} + 149m^{3}n^{4} - 73m^{2}n^{5} + 21mn^{6} - 3n^{7},\\
z &= -18m^{7} + 117m^{6}n - 254m^{5}n^{2} + 295m^{4}n^{3} - 210m^{3}n^{4} + 87m^{2}n^{5} - 18mn^{6} + n^{7},\\
u &= -18m^{7} + 69m^{6}n - 106m^{5}n^{2} + 119m^{4}n^{3} - 104m^{3}n^{4} + 61m^{2}n^{5} - 20mn^{6} + 3n^{7},\\
v &= 27m^{7} - 114m^{6}n + 241m^{5}n^{2} - 298m^{4}n^{3} + 213m^{3}n^{4} - 86m^{2}n^{5} + 19mn^{6} - 2n^{7},\\
w &= -9m^{7} + 75m^{6}n - 159m^{5}n^{2} + 155m^{4}n^{3} - 75m^{3}n^{4} + 21m^{2}n^{5} - 5mn^{6} + n^{7}.
\end{align*}