\(\text{PSU}(2)_5\times \mathbb{Z}_3:\ \text{FR}^{9,6}_{10}\)
Fusion Rules
\[\begin{array}{|lllllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{2} & \mathbf{3} & \mathbf{1} & \mathbf{6} & \mathbf{4} & \mathbf{5} & \mathbf{8} & \mathbf{9} & \mathbf{7} \\ \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{6} & \mathbf{4} & \mathbf{9} & \mathbf{7} & \mathbf{8} \\ \mathbf{4} & \mathbf{6} & \mathbf{5} & \mathbf{1}+\mathbf{7} & \mathbf{3}+\mathbf{9} & \mathbf{2}+\mathbf{8} & \mathbf{4}+\mathbf{7} & \mathbf{6}+\mathbf{8} & \mathbf{5}+\mathbf{9} \\ \mathbf{5} & \mathbf{4} & \mathbf{6} & \mathbf{3}+\mathbf{9} & \mathbf{2}+\mathbf{8} & \mathbf{1}+\mathbf{7} & \mathbf{5}+\mathbf{9} & \mathbf{4}+\mathbf{7} & \mathbf{6}+\mathbf{8} \\ \mathbf{6} & \mathbf{5} & \mathbf{4} & \mathbf{2}+\mathbf{8} & \mathbf{1}+\mathbf{7} & \mathbf{3}+\mathbf{9} & \mathbf{6}+\mathbf{8} & \mathbf{5}+\mathbf{9} & \mathbf{4}+\mathbf{7} \\ \mathbf{7} & \mathbf{8} & \mathbf{9} & \mathbf{4}+\mathbf{7} & \mathbf{5}+\mathbf{9} & \mathbf{6}+\mathbf{8} & \mathbf{1}+\mathbf{4}+\mathbf{7} & \mathbf{2}+\mathbf{6}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{9} \\ \mathbf{8} & \mathbf{9} & \mathbf{7} & \mathbf{6}+\mathbf{8} & \mathbf{4}+\mathbf{7} & \mathbf{5}+\mathbf{9} & \mathbf{2}+\mathbf{6}+\mathbf{8} & \mathbf{3}+\mathbf{5}+\mathbf{9} & \mathbf{1}+\mathbf{4}+\mathbf{7} \\ \mathbf{9} & \mathbf{7} & \mathbf{8} & \mathbf{5}+\mathbf{9} & \mathbf{6}+\mathbf{8} & \mathbf{4}+\mathbf{7} & \mathbf{3}+\mathbf{5}+\mathbf{9} & \mathbf{1}+\mathbf{4}+\mathbf{7} & \mathbf{2}+\mathbf{6}+\mathbf{8} \\ \hline \end{array}\]The fusion rules are invariant under the group generated by the following permutations:
\[\{(\mathbf{2} \ \mathbf{3}) (\mathbf{5} \ \mathbf{6}) (\mathbf{8} \ \mathbf{9})\}\]The following particles form non-trivial sub fusion rings
| Particles | SubRing |
|---|---|
| \(\{\mathbf{1},\mathbf{2},\mathbf{3}\}\) | \(\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}\) |
| \(\{\mathbf{1},\mathbf{4},\mathbf{7}\}\) | \(\text{PSU}(2)_5:\ \text{FR}^{3,0}_{3}\) |
Quantum Dimensions
| Particle | Numeric | Symbolic |
|---|---|---|
| \(\mathbf{1}\) | \(1.\) | \(1\) |
| \(\mathbf{2}\) | \(1.\) | \(1\) |
| \(\mathbf{3}\) | \(1.\) | \(1\) |
| \(\mathbf{4}\) | \(1.80194\) | \(\text{Root}\left[x^3-x^2-2 x+1,3\right]\) |
| \(\mathbf{5}\) | \(1.80194\) | \(\sqrt[3]{\text{Root}\left[x^3-4 x^2-11 x+1,3\right]}\) |
| \(\mathbf{6}\) | \(1.80194\) | \(\sqrt[3]{\text{Root}\left[x^3-4 x^2-11 x+1,3\right]}\) |
| \(\mathbf{7}\) | \(2.24698\) | \(\text{Root}\left[x^3-2 x^2-x+1,3\right]\) |
| \(\mathbf{8}\) | \(2.24698\) | \(\sqrt[3]{\text{Root}\left[x^3-11 x^2-4 x+1,3\right]}\) |
| \(\mathbf{9}\) | \(2.24698\) | \(\sqrt[3]{\text{Root}\left[x^3-11 x^2-4 x+1,3\right]}\) |
| \(\mathcal{D}_{FP}^2\) | \(27.8877\) | \(2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3\) |
Characters
The symbolic character table is the following
\[\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{4} & \mathbf{8} & \mathbf{9} & \mathbf{7} \\ \hline 1 & 1 & 1 & \text{Root}\left[x^3-x^2-2 x+1,3\right] & \text{Root}\left[x^3-x^2-2 x+1,3\right] & \text{Root}\left[x^3-x^2-2 x+1,3\right] & \text{Root}\left[x^3-2 x^2-x+1,3\right] & \text{Root}\left[x^3-2 x^2-x+1,3\right] & \text{Root}\left[x^3-2 x^2-x+1,3\right] \\ 1 & 1 & 1 & \text{Root}\left[x^3-x^2-2 x+1,2\right] & \text{Root}\left[x^3-x^2-2 x+1,2\right] & \text{Root}\left[x^3-x^2-2 x+1,2\right] & \text{Root}\left[x^3-2 x^2-x+1,1\right] & \text{Root}\left[x^3-2 x^2-x+1,1\right] & \text{Root}\left[x^3-2 x^2-x+1,1\right] \\ 1 & 1 & 1 & \text{Root}\left[x^3-x^2-2 x+1,1\right] & \text{Root}\left[x^3-x^2-2 x+1,1\right] & \text{Root}\left[x^3-x^2-2 x+1,1\right] & \text{Root}\left[x^3-2 x^2-x+1,2\right] & \text{Root}\left[x^3-2 x^2-x+1,2\right] & \text{Root}\left[x^3-2 x^2-x+1,2\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \text{Root}\left[x^6+x^5+3 x^4+5 x^2+2 x+1,1\right] & \text{Root}\left[x^6+x^5+3 x^4+5 x^2+2 x+1,2\right] & \text{Root}\left[x^3-x^2-2 x+1,2\right] & \text{Root}\left[x^6+2 x^5+5 x^4+3 x^2+x+1,6\right] & \text{Root}\left[x^6+2 x^5+5 x^4+3 x^2+x+1,5\right] & \text{Root}\left[x^3-2 x^2-x+1,1\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \text{Root}\left[x^6+x^5+3 x^4+5 x^2+2 x+1,2\right] & \text{Root}\left[x^6+x^5+3 x^4+5 x^2+2 x+1,1\right] & \text{Root}\left[x^3-x^2-2 x+1,2\right] & \text{Root}\left[x^6+2 x^5+5 x^4+3 x^2+x+1,5\right] & \text{Root}\left[x^6+2 x^5+5 x^4+3 x^2+x+1,6\right] & \text{Root}\left[x^3-2 x^2-x+1,1\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \text{Root}\left[x^6+x^5+3 x^4+5 x^2+2 x+1,5\right] & \text{Root}\left[x^6+x^5+3 x^4+5 x^2+2 x+1,6\right] & \text{Root}\left[x^3-x^2-2 x+1,1\right] & \text{Root}\left[x^6+2 x^5+5 x^4+3 x^2+x+1,2\right] & \text{Root}\left[x^6+2 x^5+5 x^4+3 x^2+x+1,1\right] & \text{Root}\left[x^3-2 x^2-x+1,2\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \text{Root}\left[x^6+x^5+3 x^4+5 x^2+2 x+1,6\right] & \text{Root}\left[x^6+x^5+3 x^4+5 x^2+2 x+1,5\right] & \text{Root}\left[x^3-x^2-2 x+1,1\right] & \text{Root}\left[x^6+2 x^5+5 x^4+3 x^2+x+1,1\right] & \text{Root}\left[x^6+2 x^5+5 x^4+3 x^2+x+1,2\right] & \text{Root}\left[x^3-2 x^2-x+1,2\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \text{Root}\left[x^6+x^5+3 x^4+5 x^2+2 x+1,4\right] & \text{Root}\left[x^6+x^5+3 x^4+5 x^2+2 x+1,3\right] & \text{Root}\left[x^3-x^2-2 x+1,3\right] & \text{Root}\left[x^6+2 x^5+5 x^4+3 x^2+x+1,4\right] & \text{Root}\left[x^6+2 x^5+5 x^4+3 x^2+x+1,3\right] & \text{Root}\left[x^3-2 x^2-x+1,3\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \text{Root}\left[x^6+x^5+3 x^4+5 x^2+2 x+1,3\right] & \text{Root}\left[x^6+x^5+3 x^4+5 x^2+2 x+1,4\right] & \text{Root}\left[x^3-x^2-2 x+1,3\right] & \text{Root}\left[x^6+2 x^5+5 x^4+3 x^2+x+1,3\right] & \text{Root}\left[x^6+2 x^5+5 x^4+3 x^2+x+1,4\right] & \text{Root}\left[x^3-2 x^2-x+1,3\right] \\ \hline \end{array}\]The numeric character table is the following
\[\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{6} & \mathbf{5} & \mathbf{4} & \mathbf{8} & \mathbf{9} & \mathbf{7} \\ \hline 1.000 & 1.000 & 1.000 & 1.802 & 1.802 & 1.802 & 2.247 & 2.247 & 2.247 \\ 1.000 & 1.000 & 1.000 & 0.4450 & 0.4450 & 0.4450 & -0.8019 & -0.8019 & -0.8019 \\ 1.000 & 1.000 & 1.000 & -1.247 & -1.247 & -1.247 & 0.5550 & 0.5550 & 0.5550 \\ 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i & -0.2225-0.3854 i & -0.2225+0.3854 i & 0.4450 & 0.4010+0.6945 i & 0.4010-0.6945 i & -0.8019 \\ 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i & -0.2225+0.3854 i & -0.2225-0.3854 i & 0.4450 & 0.4010-0.6945 i & 0.4010+0.6945 i & -0.8019 \\ 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i & 0.6235-1.0799 i & 0.6235+1.0799 i & -1.247 & -0.2775+0.4806 i & -0.2775-0.4806 i & 0.5550 \\ 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i & 0.6235+1.0799 i & 0.6235-1.0799 i & -1.247 & -0.2775-0.4806 i & -0.2775+0.4806 i & 0.5550 \\ 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i & -0.901+1.561 i & -0.901-1.561 i & 1.802 & -1.123+1.946 i & -1.123-1.946 i & 2.247 \\ 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i & -0.901-1.561 i & -0.901+1.561 i & 1.802 & -1.123-1.946 i & -1.123+1.946 i & 2.247 \\ \hline \end{array}\]Modular Data
The matching \(S\)-matrices and twist factors are the following
| \(S\)-matrix | Twist factors |
|---|---|
| \(\frac{1}{\sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3}}\left(\begin{array}{ccccccccc} \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} \\ \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,3\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,1\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,2\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} \\ \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,3\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,2\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,1\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} \\ \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,1\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,1\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,1\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} \\ \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,3\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,1\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,12\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,11\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} \\ \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,3\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,1\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,11\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,12\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} \\ \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,2\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,2\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,2\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} \\ \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,1\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,2\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,2\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,10\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,9\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} \\ \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,2\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,1\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,2\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,9\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,10\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} \\\end{array}\right)\) | \(\begin{array}{l}\left(0,-\frac{1}{3},-\frac{1}{3},\frac{1}{7},-\frac{4}{21},-\frac{4}{21},-\frac{2}{7},\frac{8}{21},\frac{8}{21}\right) \\\left(0,-\frac{1}{3},-\frac{1}{3},-\frac{1}{7},-\frac{10}{21},-\frac{10}{21},\frac{2}{7},-\frac{1}{21},-\frac{1}{21}\right)\end{array}\) |
| \(\frac{1}{\sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3}}\left(\begin{array}{ccccccccc} \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} \\ \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,3\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,2\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,1\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} \\ \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,3\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,1\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,2\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} \\ \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,1\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,1\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,1\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} \\ \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,3\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,1\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,11\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,12\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} \\ \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,3\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,1\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,12\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,11\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} \\ \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,2\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,2\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,2\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} \\ \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,2\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,1\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,2\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,9\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,10\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} \\ \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,1\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,2\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,4\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,6\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,5\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1323 x^6-441 x^4+42 x^2-1,2\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,10\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} & \text{Root}\left[1750329 x^{12}+583443 x^{10}+138915 x^8+15876 x^6+1323 x^4+42 x^2+1,9\right] \sqrt{2 \text{Root}\left[x^3-11 x^2-4 x+1,3\right]^{2/3}+\text{Root}\left[x^3-x^2-2 x+1,3\right]^2+\text{Root}\left[x^3-2 x^2-x+1,3\right]^2+2 \text{Root}\left[x^3-4 x^2-11 x+1,3\right]^{2/3}+3} \\\end{array}\right)\) | \(\begin{array}{l}\left(0,\frac{1}{3},\frac{1}{3},\frac{1}{7},\frac{10}{21},\frac{10}{21},-\frac{2}{7},\frac{1}{21},\frac{1}{21}\right) \\\left(0,\frac{1}{3},\frac{1}{3},-\frac{1}{7},\frac{4}{21},\frac{4}{21},\frac{2}{7},-\frac{8}{21},-\frac{8}{21}\right)\end{array}\) |
Adjoint Subring
Particles \(\mathbf{1}, \mathbf{4}, \mathbf{7}\), form the adjoint subring \(\text{PSU}(2)_5:\ \text{FR}^{3,0}_{3}\) .
The upper central series is the following: \(\text{PSU}(2)_5\times \mathbb{Z}_3 \underset{ \mathbf{1}, \mathbf{4}, \mathbf{7} }{\supset} \text{PSU}(2)_5\)
Universal grading
Each particle can be graded as follows: \(\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}', \text{deg}(\mathbf{3}) = \mathbf{3}', \text{deg}(\mathbf{4}) = \mathbf{1}', \text{deg}(\mathbf{5}) = \mathbf{3}', \text{deg}(\mathbf{6}) = \mathbf{2}', \text{deg}(\mathbf{7}) = \mathbf{1}', \text{deg}(\mathbf{8}) = \mathbf{2}', \text{deg}(\mathbf{9}) = \mathbf{3}'\), where the degrees form the group \(\mathbb{Z}_3\) with multiplication table:
\[\begin{array}{|lll|} \hline \mathbf{1}' & \mathbf{2}' & \mathbf{3}' \\ \mathbf{2}' & \mathbf{3}' & \mathbf{1}' \\ \mathbf{3}' & \mathbf{1}' & \mathbf{2}' \\ \hline \end{array}\]Categorifications
Data
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