\(\text{PSU}(2)_{13}:\ \text{FR}^{7,0}_{17}\)
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)\) | \(\begin{array}{l}\left(0,\frac{1}{5},-\frac{2}{15},0,-\frac{2}{5},-\frac{1}{3},\frac{1}{5}\right) \\\left(0,-\frac{1}{5},\frac{2}{15},0,\frac{2}{5},\frac{1}{3},-\frac{1}{5}\right)\end{array}\) |
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: