Basic invariants
| Dimension: | $8$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(6784487424\)\(\medspace = 2^{12} \cdot 3^{4} \cdot 11^{2} \cdot 13^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.17966327808.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T213 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_3\wr S_3$ |
| Projective stem field: | Galois closure of 9.1.17966327808.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - x^{8} + 3x^{7} + x^{6} - x^{5} - x^{4} - 4x^{2} - 3x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 211 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 211 }$:
\( x^{3} + 2x + 209 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 91\cdot 211 + 54\cdot 211^{2} + 145\cdot 211^{3} + 134\cdot 211^{4} + 142\cdot 211^{5} + 135\cdot 211^{6} + 157\cdot 211^{7} + 11\cdot 211^{8} + 68\cdot 211^{9} +O(211^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 114 + 186\cdot 211 + 82\cdot 211^{2} + 133\cdot 211^{3} + 191\cdot 211^{4} + 23\cdot 211^{5} + 180\cdot 211^{6} + 164\cdot 211^{7} + 3\cdot 211^{8} + 6\cdot 211^{9} +O(211^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 149 + 122\cdot 211 + 33\cdot 211^{2} + 77\cdot 211^{3} + 41\cdot 211^{4} + 69\cdot 211^{5} + 34\cdot 211^{6} + 202\cdot 211^{7} + 93\cdot 211^{8} + 60\cdot 211^{9} +O(211^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 21 a^{2} + 119 a + 115 + \left(207 a^{2} + 25 a + 176\right)\cdot 211 + \left(152 a^{2} + 90 a + 123\right)\cdot 211^{2} + \left(190 a^{2} + 172 a + 4\right)\cdot 211^{3} + \left(12 a^{2} + 107 a + 173\right)\cdot 211^{4} + \left(170 a^{2} + 133 a + 21\right)\cdot 211^{5} + \left(209 a^{2} + 138 a + 198\right)\cdot 211^{6} + \left(121 a^{2} + 7 a + 33\right)\cdot 211^{7} + \left(35 a^{2} + 180 a + 86\right)\cdot 211^{8} + \left(137 a^{2} + 100 a + 198\right)\cdot 211^{9} +O(211^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 28 a^{2} + 69 a + 3 + \left(90 a^{2} + 120 a + 16\right)\cdot 211 + \left(72 a^{2} + 5 a + 190\right)\cdot 211^{2} + \left(51 a^{2} + 47 a + 58\right)\cdot 211^{3} + \left(150 a^{2} + 169 a + 203\right)\cdot 211^{4} + \left(192 a^{2} + 180 a + 101\right)\cdot 211^{5} + \left(63 a^{2} + 45 a + 120\right)\cdot 211^{6} + \left(105 a^{2} + 22 a + 164\right)\cdot 211^{7} + \left(71 a^{2} + 104 a + 160\right)\cdot 211^{8} + \left(109 a^{2} + 114 a + 155\right)\cdot 211^{9} +O(211^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 47 a^{2} + 67 a + 9 + \left(140 a^{2} + 71 a + 17\right)\cdot 211 + \left(149 a^{2} + 153 a + 49\right)\cdot 211^{2} + \left(163 a^{2} + 140 a + 109\right)\cdot 211^{3} + \left(133 a^{2} + 46 a + 193\right)\cdot 211^{4} + \left(201 a^{2} + 207 a + 63\right)\cdot 211^{5} + \left(128 a^{2} + 80 a + 90\right)\cdot 211^{6} + \left(66 a^{2} + 197 a + 30\right)\cdot 211^{7} + \left(111 a^{2} + 195 a + 187\right)\cdot 211^{8} + \left(132 a^{2} + 122 a + 121\right)\cdot 211^{9} +O(211^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 58 a^{2} + 92 a + 43 + \left(5 a^{2} + 25 a + 184\right)\cdot 211 + \left(13 a^{2} + 85 a + 110\right)\cdot 211^{2} + \left(178 a^{2} + 8 a + 157\right)\cdot 211^{3} + \left(121 a^{2} + 147 a + 24\right)\cdot 211^{4} + \left(198 a^{2} + 6 a + 180\right)\cdot 211^{5} + \left(92 a^{2} + 83 a + 88\right)\cdot 211^{6} + \left(92 a^{2} + 205 a + 147\right)\cdot 211^{7} + \left(158 a^{2} + 80 a + 65\right)\cdot 211^{8} + \left(68 a^{2} + 117 a + 31\right)\cdot 211^{9} +O(211^{10})\)
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| $r_{ 8 }$ | $=$ |
\( 125 a^{2} + 50 a + 62 + \left(115 a^{2} + 65 a + 120\right)\cdot 211 + \left(125 a^{2} + 120 a + 190\right)\cdot 211^{2} + \left(192 a^{2} + 155 a + 176\right)\cdot 211^{3} + \left(149 a^{2} + 105 a + 202\right)\cdot 211^{4} + \left(30 a^{2} + 23 a + 96\right)\cdot 211^{5} + \left(54 a^{2} + 82 a + 107\right)\cdot 211^{6} + \left(13 a^{2} + 194 a + 182\right)\cdot 211^{7} + \left(192 a^{2} + 25 a + 180\right)\cdot 211^{8} + \left(32 a^{2} + 190 a + 53\right)\cdot 211^{9} +O(211^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 143 a^{2} + 25 a + 137 + \left(74 a^{2} + 114 a + 140\right)\cdot 211 + \left(119 a^{2} + 178 a + 8\right)\cdot 211^{2} + \left(67 a^{2} + 108 a + 192\right)\cdot 211^{3} + \left(64 a^{2} + 56 a + 100\right)\cdot 211^{4} + \left(50 a^{2} + 81 a + 143\right)\cdot 211^{5} + \left(83 a^{2} + 202 a + 99\right)\cdot 211^{6} + \left(22 a^{2} + 5 a + 182\right)\cdot 211^{7} + \left(64 a^{2} + 46 a + 53\right)\cdot 211^{8} + \left(152 a^{2} + 198 a + 148\right)\cdot 211^{9} +O(211^{10})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 9 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 9 }$ | Character value |
| $1$ | $1$ | $()$ | $8$ |
| $9$ | $2$ | $(1,2)$ | $0$ |
| $18$ | $2$ | $(1,4)(2,6)(3,9)$ | $4$ |
| $27$ | $2$ | $(1,2)(4,6)(5,7)$ | $0$ |
| $27$ | $2$ | $(1,2)(5,7)$ | $0$ |
| $54$ | $2$ | $(1,2)(4,5)(6,7)(8,9)$ | $0$ |
| $6$ | $3$ | $(5,7,8)$ | $-4$ |
| $8$ | $3$ | $(1,3,2)(4,9,6)(5,8,7)$ | $-1$ |
| $12$ | $3$ | $(4,6,9)(5,7,8)$ | $2$ |
| $72$ | $3$ | $(1,4,5)(2,6,7)(3,9,8)$ | $2$ |
| $54$ | $4$ | $(1,5,2,7)(3,8)$ | $0$ |
| $162$ | $4$ | $(1,5,2,7)(3,8)(6,9)$ | $0$ |
| $36$ | $6$ | $(1,4)(2,6)(3,9)(5,7,8)$ | $-2$ |
| $36$ | $6$ | $(1,5,3,8,2,7)$ | $-2$ |
| $36$ | $6$ | $(1,2)(5,7,8)$ | $0$ |
| $36$ | $6$ | $(1,2)(4,6,9)(5,7,8)$ | $0$ |
| $54$ | $6$ | $(1,2)(4,9,6)(5,7)$ | $0$ |
| $72$ | $6$ | $(1,4,2,6,3,9)(5,7,8)$ | $1$ |
| $108$ | $6$ | $(1,2)(4,5,6,7,9,8)$ | $0$ |
| $216$ | $6$ | $(1,4,5,2,6,7)(3,9,8)$ | $0$ |
| $144$ | $9$ | $(1,4,5,3,9,8,2,6,7)$ | $-1$ |
| $108$ | $12$ | $(1,5,2,7)(3,8)(4,6,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.