\(\text{PSU}(2)_{13}:\ \text{FR}^{7,0}_{17}\)

Fusion Rules

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

Quantum Dimensions

Particle	Numeric	Symbolic
\(\mathbf{1}\)	\(1.\)	\(1\)
\(\mathbf{2}\)	\(1.9563\)	\(\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]\)
\(\mathbf{3}\)	\(2.82709\)	\(\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]\)
\(\mathbf{4}\)	\(3.57433\)	\(\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]\)
\(\mathbf{5}\)	\(4.16535\)	\(\frac{1}{2} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)\)
\(\mathbf{6}\)	\(4.57433\)	\(\text{Root}\left[x^4-5 x^3+10 x-5,4\right]\)
\(\mathbf{7}\)	\(4.78339\)	\(\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]\)
\(\mathcal{D}_{FP}^2\)	\(86.7508\)	\(\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2\)

Characters

The symbolic character table is the following

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

The numeric character table is the following

\[\begin{array}{|rrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\ \hline 1.000 & 1.956 & 2.827 & 3.574 & 4.165 & 4.574 & 4.783 \\ 1.000 & 1.618 & 1.618 & 1.000 & 0 & -1.000 & -1.618 \\ 1.000 & 1.000 & 0 & -1.000 & -1.000 & 0 & 1.000 \\ 1.000 & 0.2091 & -0.9563 & -0.4090 & 0.8708 & 0.5910 & -0.7472 \\ 1.000 & -0.6180 & -0.6180 & 1.000 & 0 & -1.000 & 0.6180 \\ 1.000 & -1.338 & 0.7909 & 0.2798 & -1.165 & 1.280 & -0.5473 \\ 1.000 & -1.827 & 2.338 & -2.445 & 2.129 & -1.445 & 0.5112 \\ \hline \end{array}\]

Modular Data

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

\(S\)-matrix	Twist factors
\(\frac{1}{\sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2}}\left(\begin{array}{ccccccc} \text{Root}\left[225 x^4+225 x^3+60 x^2-1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4-225 x^3+60 x^2-1,2\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,3\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4-225 x^3+60 x^2-1,3\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \frac{1}{\sqrt{\frac{5}{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2}}} & \text{Root}\left[45 x^4-15 x^2+1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4-225 x^3+60 x^2-1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} \\ \text{Root}\left[225 x^4-225 x^3+60 x^2-1,2\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4+225 x^3+60 x^2-1,2\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4+225 x^3+60 x^2-1,1\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \frac{1}{\sqrt{\frac{5}{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2}}} & \text{Root}\left[45 x^4-15 x^2+1,2\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4+225 x^3+60 x^2-1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} \\ \text{Root}\left[45 x^4-15 x^2+1,3\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,3\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & 0 & \text{Root}\left[45 x^4-15 x^2+1,2\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,1\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} \\ \text{Root}\left[225 x^4-225 x^3+60 x^2-1,3\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4+225 x^3+60 x^2-1,1\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,3\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4+225 x^3+60 x^2-1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & -\frac{1}{\sqrt{\frac{5}{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2}}} & \text{Root}\left[45 x^4-15 x^2+1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4+225 x^3+60 x^2-1,3\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} \\ \frac{1}{\sqrt{\frac{5}{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2}}} & \frac{1}{\sqrt{\frac{5}{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2}}} & 0 & -\frac{1}{\sqrt{\frac{5}{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2}}} & -\frac{1}{\sqrt{\frac{5}{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2}}} & 0 & \frac{1}{\sqrt{\frac{5}{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2}}} \\ \text{Root}\left[45 x^4-15 x^2+1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,2\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,2\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & 0 & \text{Root}\left[45 x^4-15 x^2+1,1\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,3\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} \\ \text{Root}\left[225 x^4-225 x^3+60 x^2-1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4+225 x^3+60 x^2-1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,1\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4+225 x^3+60 x^2-1,3\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \frac{1}{\sqrt{\frac{5}{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2}}} & \text{Root}\left[45 x^4-15 x^2+1,3\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4+225 x^3+60 x^2-1,2\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} \\\end{array}\right)\)	\(\begin{array}{l}\left(0,\frac{1}{5},-\frac{2}{15},0,-\frac{2}{5},-\frac{1}{3},\frac{1}{5}\right) \\\left(0,-\frac{1}{5},\frac{2}{15},0,\frac{2}{5},\frac{1}{3},-\frac{1}{5}\right)\end{array}\)

\(S\)-matrix

Twist factors

\(\frac{1}{\sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2}}\left(\begin{array}{ccccccc} \text{Root}\left[225 x^4+225 x^3+60 x^2-1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4-225 x^3+60 x^2-1,2\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,3\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4-225 x^3+60 x^2-1,3\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \frac{1}{\sqrt{\frac{5}{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2}}} & \text{Root}\left[45 x^4-15 x^2+1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4-225 x^3+60 x^2-1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} \\ \text{Root}\left[225 x^4-225 x^3+60 x^2-1,2\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4+225 x^3+60 x^2-1,2\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4+225 x^3+60 x^2-1,1\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \frac{1}{\sqrt{\frac{5}{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2}}} & \text{Root}\left[45 x^4-15 x^2+1,2\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4+225 x^3+60 x^2-1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} \\ \text{Root}\left[45 x^4-15 x^2+1,3\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,3\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & 0 & \text{Root}\left[45 x^4-15 x^2+1,2\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,1\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} \\ \text{Root}\left[225 x^4-225 x^3+60 x^2-1,3\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4+225 x^3+60 x^2-1,1\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,3\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4+225 x^3+60 x^2-1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & -\frac{1}{\sqrt{\frac{5}{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2}}} & \text{Root}\left[45 x^4-15 x^2+1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4+225 x^3+60 x^2-1,3\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} \\ \frac{1}{\sqrt{\frac{5}{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2}}} & \frac{1}{\sqrt{\frac{5}{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2}}} & 0 & -\frac{1}{\sqrt{\frac{5}{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2}}} & -\frac{1}{\sqrt{\frac{5}{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2}}} & 0 & \frac{1}{\sqrt{\frac{5}{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2}}} \\ \text{Root}\left[45 x^4-15 x^2+1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,2\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,2\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & 0 & \text{Root}\left[45 x^4-15 x^2+1,1\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,3\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} \\ \text{Root}\left[225 x^4-225 x^3+60 x^2-1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4+225 x^3+60 x^2-1,4\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[45 x^4-15 x^2+1,1\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4+225 x^3+60 x^2-1,3\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \frac{1}{\sqrt{\frac{5}{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2}}} & \text{Root}\left[45 x^4-15 x^2+1,3\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} & \text{Root}\left[225 x^4+225 x^3+60 x^2-1,2\right] \sqrt{\text{Root}\left[x^4-5 x^3+10 x-5,4\right]^2+\text{Root}\left[x^4+x^3-4 x^2-4 x+1,4\right]^2+\text{Root}\left[x^4-5 x^3+5 x^2+5 x-5,4\right]^2+\text{Root}\left[x^4-x^3-9 x^2-x+1,4\right]^2+\text{Root}\left[x^4-4 x^3-4 x^2+x+1,4\right]^2+1+\frac{1}{4} \left(3+\sqrt{3 \left(5+2 \sqrt{5}\right)}\right)^2} \\\end{array}\right)\)

\(\begin{array}{l}\left(0,\frac{1}{5},-\frac{2}{15},0,-\frac{2}{5},-\frac{1}{3},\frac{1}{5}\right) \\\left(0,-\frac{1}{5},\frac{2}{15},0,\frac{2}{5},\frac{1}{3},-\frac{1}{5}\right)\end{array}\)

Adjoint Subring

The adjoint subring is the ring itself.

The upper central series is trivial.

Universal grading

This fusion ring allows only the trivial grading.

Categorifications

Data

Download links for numeric data:

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