\(\text{SU}(2)_8:\ \text{FR}^{9,0}_{27}\)

Fusion Rules

\[\begin{array}{|lllllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} & \mathbf{9} \\ \mathbf{3} & \mathbf{4} & \mathbf{1}+\mathbf{6} & \mathbf{2}+\mathbf{5} & \mathbf{4}+\mathbf{8} & \mathbf{3}+\mathbf{7} & \mathbf{6}+\mathbf{9} & \mathbf{5}+\mathbf{9} & \mathbf{7}+\mathbf{8} \\ \mathbf{4} & \mathbf{3} & \mathbf{2}+\mathbf{5} & \mathbf{1}+\mathbf{6} & \mathbf{3}+\mathbf{7} & \mathbf{4}+\mathbf{8} & \mathbf{5}+\mathbf{9} & \mathbf{6}+\mathbf{9} & \mathbf{7}+\mathbf{8} \\ \mathbf{5} & \mathbf{6} & \mathbf{4}+\mathbf{8} & \mathbf{3}+\mathbf{7} & \mathbf{1}+\mathbf{6}+\mathbf{9} & \mathbf{2}+\mathbf{5}+\mathbf{9} & \mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{9} \\ \mathbf{6} & \mathbf{5} & \mathbf{3}+\mathbf{7} & \mathbf{4}+\mathbf{8} & \mathbf{2}+\mathbf{5}+\mathbf{9} & \mathbf{1}+\mathbf{6}+\mathbf{9} & \mathbf{3}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{9} \\ \mathbf{7} & \mathbf{8} & \mathbf{6}+\mathbf{9} & \mathbf{5}+\mathbf{9} & \mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8} \\ \mathbf{8} & \mathbf{7} & \mathbf{5}+\mathbf{9} & \mathbf{6}+\mathbf{9} & \mathbf{3}+\mathbf{7}+\mathbf{8} & \mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{1}+\mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8} \\ \mathbf{9} & \mathbf{9} & \mathbf{7}+\mathbf{8} & \mathbf{7}+\mathbf{8} & \mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{5}+\mathbf{6}+\mathbf{9} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{3}+\mathbf{4}+\mathbf{7}+\mathbf{8} & \mathbf{1}+\mathbf{2}+\mathbf{5}+\mathbf{6}+\mathbf{9} \\ \hline \end{array}\]

The fusion rules are invariant under the group generated by the following permutations:

\[\{(\mathbf{3} \ \mathbf{4}) (\mathbf{7} \ \mathbf{8})\}\]

The following particles form non-trivial sub fusion rings

Particles	SubRing
\(\{\mathbf{1},\mathbf{2}\}\)	\(\mathbb{Z}_2:\ \text{FR}^{2,0}_{1}\)
\(\{\mathbf{1},\mathbf{2},\mathbf{5},\mathbf{6},\mathbf{9}\}\)	\(\text{PSU(2})_8:\ \text{FR}^{5,0}_{7}\)

Quantum Dimensions

Particle	Numeric	Symbolic
\(\mathbf{1}\)	\(1.\)	\(1\)
\(\mathbf{2}\)	\(1.\)	\(1\)
\(\mathbf{3}\)	\(1.90211\)	\(\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}\)
\(\mathbf{4}\)	\(1.90211\)	\(\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}\)
\(\mathbf{5}\)	\(2.61803\)	\(\frac{1}{2} \left(3+\sqrt{5}\right)\)
\(\mathbf{6}\)	\(2.61803\)	\(\frac{1}{2} \left(3+\sqrt{5}\right)\)
\(\mathbf{7}\)	\(3.07768\)	\(\sqrt{5+2 \sqrt{5}}\)
\(\mathbf{8}\)	\(3.07768\)	\(\sqrt{5+2 \sqrt{5}}\)
\(\mathbf{9}\)	\(3.23607\)	\(1+\sqrt{5}\)
\(\mathcal{D}_{FP}^2\)	\(52.3607\)	\(17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2\)

Characters

The symbolic character table is the following

\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} & \mathbf{9} \\ \hline 1 & 1 & \text{Root}\left[x^4-5 x^2+5,4\right] & \text{Root}\left[x^4-5 x^2+5,4\right] & \frac{1}{2} \left(3+\sqrt{5}\right) & \frac{1}{2} \left(3+\sqrt{5}\right) & \text{Root}\left[x^4-10 x^2+5,4\right] & \text{Root}\left[x^4-10 x^2+5,4\right] & 1+\sqrt{5} \\ 1 & 1 & \text{Root}\left[x^4-5 x^2+5,3\right] & \text{Root}\left[x^4-5 x^2+5,3\right] & \frac{1}{2} \left(3-\sqrt{5}\right) & \frac{1}{2} \left(3-\sqrt{5}\right) & \text{Root}\left[x^4-10 x^2+5,2\right] & \text{Root}\left[x^4-10 x^2+5,2\right] & 1-\sqrt{5} \\ 1 & 1 & 0 & 0 & -1 & -1 & 0 & 0 & 1 \\ 1 & 1 & \text{Root}\left[x^4-5 x^2+5,2\right] & \text{Root}\left[x^4-5 x^2+5,2\right] & \frac{1}{2} \left(3-\sqrt{5}\right) & \frac{1}{2} \left(3-\sqrt{5}\right) & \text{Root}\left[x^4-10 x^2+5,3\right] & \text{Root}\left[x^4-10 x^2+5,3\right] & 1-\sqrt{5} \\ 1 & 1 & \text{Root}\left[x^4-5 x^2+5,1\right] & \text{Root}\left[x^4-5 x^2+5,1\right] & \frac{1}{2} \left(3+\sqrt{5}\right) & \frac{1}{2} \left(3+\sqrt{5}\right) & \text{Root}\left[x^4-10 x^2+5,1\right] & \text{Root}\left[x^4-10 x^2+5,1\right] & 1+\sqrt{5} \\ 1 & -1 & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & -1 & 1 & 0 \\ 1 & -1 & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & 1 & -1 & 0 \\ 1 & -1 & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & 1 & -1 & 0 \\ 1 & -1 & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & \frac{1}{2} \left(1-\sqrt{5}\right) & \frac{1}{2} \left(\sqrt{5}-1\right) & -1 & 1 & 0 \\ \hline \end{array}\]

The numeric character table is the following

\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{6} & \mathbf{5} & \mathbf{8} & \mathbf{7} & \mathbf{9} \\ \hline 1.000 & 1.000 & 1.902 & 1.902 & 2.618 & 2.618 & 3.078 & 3.078 & 3.236 \\ 1.000 & 1.000 & 1.176 & 1.176 & 0.3820 & 0.3820 & -0.7265 & -0.7265 & -1.236 \\ 1.000 & 1.000 & 0 & 0 & -1.000 & -1.000 & 0 & 0 & 1.000 \\ 1.000 & 1.000 & -1.176 & -1.176 & 0.3820 & 0.3820 & 0.7265 & 0.7265 & -1.236 \\ 1.000 & 1.000 & -1.902 & -1.902 & 2.618 & 2.618 & -3.078 & -3.078 & 3.236 \\ 1.000 & -1.000 & 1.618 & -1.618 & 1.618 & -1.618 & -1.000 & 1.000 & 0 \\ 1.000 & -1.000 & -1.618 & 1.618 & 1.618 & -1.618 & 1.000 & -1.000 & 0 \\ 1.000 & -1.000 & 0.6180 & -0.6180 & -0.6180 & 0.6180 & 1.000 & -1.000 & 0 \\ 1.000 & -1.000 & -0.6180 & 0.6180 & -0.6180 & 0.6180 & -1.000 & 1.000 & 0 \\ \hline \end{array}\]

Modular Data

The matching \(S\)-matrices and twist factors are the following

\(S\)-matrix	Twist factors
\(\frac{1}{\sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}\left(\begin{array}{ccccccccc} \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} \\ \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} \\ \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & 0 \\ \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & 0 \\ \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & -\frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} \\ \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & -\frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} \\ \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & 0 \\ \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & 0 \\ \frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} & \frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} & 0 & 0 & -\frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} & -\frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} & 0 & 0 & \frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} \\\end{array}\right)\)	\(\begin{array}{l}\left(0,0,-\frac{3}{40},\frac{17}{40},-\frac{1}{5},-\frac{1}{5},-\frac{3}{8},\frac{1}{8},\frac{2}{5}\right) \\\left(0,0,\frac{17}{40},-\frac{3}{40},-\frac{1}{5},-\frac{1}{5},\frac{1}{8},-\frac{3}{8},\frac{2}{5}\right) \\\left(0,0,-\frac{17}{40},\frac{3}{40},\frac{1}{5},\frac{1}{5},-\frac{1}{8},\frac{3}{8},-\frac{2}{5}\right) \\\left(0,0,\frac{3}{40},-\frac{17}{40},\frac{1}{5},\frac{1}{5},\frac{3}{8},-\frac{1}{8},-\frac{2}{5}\right)\end{array}\)
\(\frac{1}{\sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}\left(\begin{array}{ccccccccc} \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} \\ \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} \\ \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & 0 \\ \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & 0 \\ \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & -\frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} \\ \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & -\frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} \\ \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & 0 \\ \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & 0 \\ \frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} & \frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} & 0 & 0 & -\frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} & -\frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} & 0 & 0 & \frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} \\\end{array}\right)\)	\(\begin{array}{l}\left(0,0,\frac{13}{40},-\frac{7}{40},\frac{1}{5},\frac{1}{5},-\frac{3}{8},\frac{1}{8},-\frac{2}{5}\right) \\\left(0,0,-\frac{7}{40},\frac{13}{40},\frac{1}{5},\frac{1}{5},\frac{1}{8},-\frac{3}{8},-\frac{2}{5}\right) \\\left(0,0,\frac{7}{40},-\frac{13}{40},-\frac{1}{5},-\frac{1}{5},-\frac{1}{8},\frac{3}{8},\frac{2}{5}\right) \\\left(0,0,-\frac{13}{40},\frac{7}{40},-\frac{1}{5},-\frac{1}{5},\frac{3}{8},-\frac{1}{8},\frac{2}{5}\right)\end{array}\)

\(S\)-matrix

Twist factors

\(\frac{1}{\sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}\left(\begin{array}{ccccccccc} \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} \\ \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} \\ \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & 0 \\ \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & 0 \\ \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & -\frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} \\ \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & -\frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} \\ \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & 0 \\ \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & 0 \\ \frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} & \frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} & 0 & 0 & -\frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} & -\frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} & 0 & 0 & \frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} \\\end{array}\right)\)

\(\begin{array}{l}\left(0,0,-\frac{3}{40},\frac{17}{40},-\frac{1}{5},-\frac{1}{5},-\frac{3}{8},\frac{1}{8},\frac{2}{5}\right) \\\left(0,0,\frac{17}{40},-\frac{3}{40},-\frac{1}{5},-\frac{1}{5},\frac{1}{8},-\frac{3}{8},\frac{2}{5}\right) \\\left(0,0,-\frac{17}{40},\frac{3}{40},\frac{1}{5},\frac{1}{5},-\frac{1}{8},\frac{3}{8},-\frac{2}{5}\right) \\\left(0,0,\frac{3}{40},-\frac{17}{40},\frac{1}{5},\frac{1}{5},\frac{3}{8},-\frac{1}{8},-\frac{2}{5}\right)\end{array}\)

\(\frac{1}{\sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}\left(\begin{array}{ccccccccc} \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} \\ \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} \\ \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & 0 \\ \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & 0 \\ \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & -\frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} \\ \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5+\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \frac{1}{20} \left(5-\sqrt{5}\right) \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & -\frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} \\ \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & 0 \\ \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,3\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,2\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,1\right] & \sqrt{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2} \text{Root}\left[80 x^4-20 x^2+1,4\right] & 0 \\ \frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} & \frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} & 0 & 0 & -\frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} & -\frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} & 0 & 0 & \frac{1}{\sqrt{\frac{5}{17+5 \sqrt{5}+\left(1+\sqrt{5}\right)^2+\frac{1}{2} \left(3+\sqrt{5}\right)^2}}} \\\end{array}\right)\)

\(\begin{array}{l}\left(0,0,\frac{13}{40},-\frac{7}{40},\frac{1}{5},\frac{1}{5},-\frac{3}{8},\frac{1}{8},-\frac{2}{5}\right) \\\left(0,0,-\frac{7}{40},\frac{13}{40},\frac{1}{5},\frac{1}{5},\frac{1}{8},-\frac{3}{8},-\frac{2}{5}\right) \\\left(0,0,\frac{7}{40},-\frac{13}{40},-\frac{1}{5},-\frac{1}{5},-\frac{1}{8},\frac{3}{8},\frac{2}{5}\right) \\\left(0,0,-\frac{13}{40},\frac{7}{40},-\frac{1}{5},-\frac{1}{5},\frac{3}{8},-\frac{1}{8},\frac{2}{5}\right)\end{array}\)

Adjoint Subring

Particles \(\mathbf{1}, \mathbf{2}, \mathbf{5}, \mathbf{6}, \mathbf{9}\), form the adjoint subring \(\text{PSU(2})_8:\ \text{FR}^{5,0}_{7}\) .

The upper central series is the following: \(\text{SU}(2)_8 \underset{ \mathbf{1}, \mathbf{2}, \mathbf{5}, \mathbf{6}, \mathbf{9} }{\supset} \text{PSU(2})_8\)

Universal grading

Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{1}', \text{deg}(\mathbf{3}) = \mathbf{2}', \text{deg}(\mathbf{4}) = \mathbf{2}', \text{deg}(\mathbf{5}) = \mathbf{1}', \text{deg}(\mathbf{6}) = \mathbf{1}', \text{deg}(\mathbf{7}) = \mathbf{2}', \text{deg}(\mathbf{8}) = \mathbf{2}', \text{deg}(\mathbf{9}) = \mathbf{1}'\), where the degrees form the group \(\mathbb{Z}_2\) with multiplication table:

\[\begin{array}{|ll|} \hline \mathbf{1}' & \mathbf{2}' \\ \mathbf{2}' & \mathbf{1}' \\ \hline \end{array}\]

Categorifications

Data

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