Basic invariants
| Dimension: | $8$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(2613254400\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 5^{2} \cdot 71^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.34359456000.1 |
| 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 9.1.34359456000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - x^{8} + 4x^{7} - 6x^{6} + 8x^{5} - 2x^{4} + 4x^{2} - x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 163 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 163 }$:
\( x^{3} + 7x + 161 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 17 a^{2} + 128 a + 45 + \left(53 a^{2} + 24 a + 99\right)\cdot 163 + \left(114 a^{2} + 89 a + 1\right)\cdot 163^{2} + \left(59 a^{2} + 129 a + 36\right)\cdot 163^{3} + \left(109 a^{2} + 27 a + 50\right)\cdot 163^{4} + \left(3 a^{2} + 110 a + 12\right)\cdot 163^{5} + \left(151 a^{2} + 13 a + 114\right)\cdot 163^{6} + \left(162 a^{2} + 48 a + 69\right)\cdot 163^{7} + \left(152 a^{2} + 47 a + 28\right)\cdot 163^{8} + \left(2 a^{2} + 154 a + 21\right)\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 32 a^{2} + 21 a + 122 + \left(53 a^{2} + 32 a + 87\right)\cdot 163 + \left(97 a^{2} + 109 a + 162\right)\cdot 163^{2} + \left(88 a^{2} + 62 a + 88\right)\cdot 163^{3} + \left(148 a^{2} + 59 a + 52\right)\cdot 163^{4} + \left(50 a^{2} + 126 a + 31\right)\cdot 163^{5} + \left(111 a^{2} + 137 a + 16\right)\cdot 163^{6} + \left(36 a^{2} + 58 a + 3\right)\cdot 163^{7} + \left(40 a^{2} + 75 a + 94\right)\cdot 163^{8} + \left(146 a^{2} + 133 a + 133\right)\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 34 a^{2} + 81 a + 70 + \left(55 a^{2} + 75 a\right)\cdot 163 + \left(68 a^{2} + 72 a + 113\right)\cdot 163^{2} + \left(104 a^{2} + 116 a + 81\right)\cdot 163^{3} + \left(85 a^{2} + 48 a + 102\right)\cdot 163^{4} + \left(97 a^{2} + 81 a + 15\right)\cdot 163^{5} + \left(21 a^{2} + 141 a + 162\right)\cdot 163^{6} + \left(78 a^{2} + 81 a + 162\right)\cdot 163^{7} + \left(50 a^{2} + 4 a + 38\right)\cdot 163^{8} + \left(47 a^{2} + 124 a + 65\right)\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 34 a^{2} + 88 a + 77 + \left(100 a^{2} + 83 a + 35\right)\cdot 163 + \left(127 a^{2} + 40 a + 32\right)\cdot 163^{2} + \left(27 a^{2} + 138 a + 131\right)\cdot 163^{3} + \left(74 a^{2} + 88 a + 85\right)\cdot 163^{4} + \left(31 a^{2} + 23 a + 103\right)\cdot 163^{5} + \left(92 a^{2} + 26 a + 35\right)\cdot 163^{6} + \left(134 a^{2} + 37 a + 25\right)\cdot 163^{7} + \left(23 a^{2} + 155 a + 126\right)\cdot 163^{8} + \left(77 a^{2} + 96 a + 82\right)\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 40 a^{2} + 8 a + 140 + \left(68 a^{2} + 44 a + 29\right)\cdot 163 + \left(39 a^{2} + 111 a + 138\right)\cdot 163^{2} + \left(117 a^{2} + 11 a + 81\right)\cdot 163^{3} + \left(147 a^{2} + 122 a + 51\right)\cdot 163^{4} + \left(155 a^{2} + 78 a + 69\right)\cdot 163^{5} + \left(96 a^{2} + 152 a + 79\right)\cdot 163^{6} + \left(99 a^{2} + 38 a + 19\right)\cdot 163^{7} + \left(3 a^{2} + 52 a + 144\right)\cdot 163^{8} + \left(25 a^{2} + 88 a + 114\right)\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 97 a^{2} + 54 a + 45 + \left(9 a^{2} + 47 a + 47\right)\cdot 163 + \left(101 a^{2} + 13 a + 71\right)\cdot 163^{2} + \left(46 a^{2} + 125 a + 110\right)\cdot 163^{3} + \left(103 a^{2} + 14 a + 58\right)\cdot 163^{4} + \left(80 a^{2} + 13 a + 61\right)\cdot 163^{5} + \left(122 a^{2} + 162 a + 68\right)\cdot 163^{6} + \left(154 a^{2} + 66 a + 119\right)\cdot 163^{7} + \left(98 a^{2} + 95 a + 150\right)\cdot 163^{8} + \left(102 a^{2} + 95 a + 92\right)\cdot 163^{9} +O(163^{10})\)
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| $r_{ 7 }$ | $=$ |
\( 112 a^{2} + 117 a + 108 + \left(54 a^{2} + 62 a + 106\right)\cdot 163 + \left(143 a^{2} + a + 82\right)\cdot 163^{2} + \left(161 a^{2} + 80 a + 132\right)\cdot 163^{3} + \left(130 a^{2} + 86 a + 96\right)\cdot 163^{4} + \left(61 a^{2} + 134 a + 120\right)\cdot 163^{5} + \left(153 a^{2} + 7 a + 70\right)\cdot 163^{6} + \left(84 a^{2} + 33 a + 140\right)\cdot 163^{7} + \left(122 a^{2} + 111 a + 157\right)\cdot 163^{8} + \left(112 a^{2} + 47 a + 98\right)\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 125 a^{2} + 105 a + 102 + \left(45 a^{2} + 141 a + 33\right)\cdot 163 + \left(6 a^{2} + 68 a + 92\right)\cdot 163^{2} + \left(126 a^{2} + 84 a + 68\right)\cdot 163^{3} + \left(70 a^{2} + 99 a + 18\right)\cdot 163^{4} + \left(31 a^{2} + 103 a + 86\right)\cdot 163^{5} + \left(85 a^{2} + 156 a + 24\right)\cdot 163^{6} + \left(52 a^{2} + 7 a + 17\right)\cdot 163^{7} + \left(29 a^{2} + 124 a + 101\right)\cdot 163^{8} + \left(19 a^{2} + 155 a + 87\right)\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 161 a^{2} + 50 a + 107 + \left(48 a^{2} + 140 a + 48\right)\cdot 163 + \left(117 a^{2} + 145 a + 121\right)\cdot 163^{2} + \left(82 a^{2} + 66 a + 83\right)\cdot 163^{3} + \left(107 a^{2} + 104 a + 135\right)\cdot 163^{4} + \left(138 a^{2} + 143 a + 151\right)\cdot 163^{5} + \left(143 a^{2} + 16 a + 80\right)\cdot 163^{6} + \left(10 a^{2} + 116 a + 94\right)\cdot 163^{7} + \left(130 a^{2} + 149 a + 136\right)\cdot 163^{8} + \left(118 a^{2} + 81 a + 117\right)\cdot 163^{9} +O(163^{10})\)
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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 |
| $1$ | $1$ | $()$ | $8$ |
| $9$ | $2$ | $(2,5)$ | $0$ |
| $18$ | $2$ | $(2,3)(4,5)(7,8)$ | $4$ |
| $27$ | $2$ | $(1,6)(2,5)(3,4)$ | $0$ |
| $27$ | $2$ | $(1,6)(2,5)$ | $0$ |
| $54$ | $2$ | $(1,3)(2,5)(4,6)(8,9)$ | $0$ |
| $6$ | $3$ | $(1,6,9)$ | $-4$ |
| $8$ | $3$ | $(1,9,6)(2,7,5)(3,8,4)$ | $-1$ |
| $12$ | $3$ | $(1,6,9)(3,4,8)$ | $2$ |
| $72$ | $3$ | $(1,2,3)(4,6,5)(7,8,9)$ | $2$ |
| $54$ | $4$ | $(1,5,6,2)(7,9)$ | $0$ |
| $162$ | $4$ | $(1,5,6,2)(4,8)(7,9)$ | $0$ |
| $36$ | $6$ | $(1,6,9)(2,3)(4,5)(7,8)$ | $-2$ |
| $36$ | $6$ | $(1,7,9,5,6,2)$ | $-2$ |
| $36$ | $6$ | $(1,6,9)(2,5)$ | $0$ |
| $36$ | $6$ | $(1,6,9)(2,5)(3,4,8)$ | $0$ |
| $54$ | $6$ | $(1,6)(2,5)(3,8,4)$ | $0$ |
| $72$ | $6$ | $(1,6,9)(2,3,5,4,7,8)$ | $1$ |
| $108$ | $6$ | $(1,4,6,8,9,3)(2,5)$ | $0$ |
| $216$ | $6$ | $(1,5,4,6,2,3)(7,8,9)$ | $0$ |
| $144$ | $9$ | $(1,7,8,9,5,4,6,2,3)$ | $-1$ |
| $108$ | $12$ | $(1,5,6,2)(3,4,8)(7,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.