Basic invariants
| Dimension: | $8$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(11757498624\)\(\medspace = 2^{8} \cdot 3^{6} \cdot 251^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.1311614290944.1 |
| 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 9.1.1311614290944.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 4x^{8} + 11x^{7} - 34x^{6} + 71x^{5} - 104x^{4} + 117x^{3} - 82x^{2} + 60x - 24 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 8 a^{2} + 51 a + 62 + \left(130 a^{2} + 119 a + 29\right)\cdot 163 + \left(46 a^{2} + 103 a + 121\right)\cdot 163^{2} + \left(161 a^{2} + 99 a + 117\right)\cdot 163^{3} + \left(128 a^{2} + 100 a + 89\right)\cdot 163^{4} + \left(151 a^{2} + 7 a + 146\right)\cdot 163^{5} + \left(125 a^{2} + 158 a + 73\right)\cdot 163^{6} + \left(81 a^{2} + 110 a + 56\right)\cdot 163^{7} + \left(87 a^{2} + 144 a + 86\right)\cdot 163^{8} + \left(140 a^{2} + 133 a + 40\right)\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 a^{2} + 157 a + 73 + \left(5 a^{2} + 136 a + 57\right)\cdot 163 + \left(80 a^{2} + 25 a + 30\right)\cdot 163^{2} + \left(52 a^{2} + 129 a + 50\right)\cdot 163^{3} + \left(128 a^{2} + 5 a + 137\right)\cdot 163^{4} + \left(42 a^{2} + 56 a + 22\right)\cdot 163^{5} + \left(24 a^{2} + 138 a + 75\right)\cdot 163^{6} + \left(17 a^{2} + 160 a\right)\cdot 163^{7} + \left(51 a^{2} + 121 a + 133\right)\cdot 163^{8} + \left(99 a^{2} + 144 a + 140\right)\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 a^{2} + 161 a + 25 + \left(80 a^{2} + 109 a + 48\right)\cdot 163 + \left(142 a^{2} + 143 a + 127\right)\cdot 163^{2} + \left(151 a^{2} + 136 a + 125\right)\cdot 163^{3} + \left(18 a^{2} + 89 a + 29\right)\cdot 163^{4} + \left(103 a^{2} + 120 a + 133\right)\cdot 163^{5} + \left(22 a^{2} + 139 a + 59\right)\cdot 163^{6} + \left(30 a^{2} + 5 a + 56\right)\cdot 163^{7} + \left(41 a^{2} + 76 a + 76\right)\cdot 163^{8} + \left(161 a^{2} + 54 a + 60\right)\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 40 a^{2} + 82 a + 137 + \left(94 a^{2} + 120 a + 59\right)\cdot 163 + \left(54 a^{2} + 53 a + 97\right)\cdot 163^{2} + \left(41 a^{2} + 36 a + 44\right)\cdot 163^{3} + \left(31 a^{2} + 105 a + 87\right)\cdot 163^{4} + \left(137 a^{2} + 149 a + 20\right)\cdot 163^{5} + \left(47 a^{2} + 17 a + 123\right)\cdot 163^{6} + \left(2 a^{2} + 3 a + 143\right)\cdot 163^{7} + \left(26 a^{2} + 80 a + 59\right)\cdot 163^{8} + \left(10 a^{2} + 26 a + 7\right)\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 54 a^{2} + 39 a + 5 + \left(91 a^{2} + 36 a + 66\right)\cdot 163 + \left(141 a^{2} + 130 a + 74\right)\cdot 163^{2} + \left(24 a^{2} + 109 a + 24\right)\cdot 163^{3} + \left(33 a^{2} + 66 a + 77\right)\cdot 163^{4} + \left(27 a^{2} + 135 a + 108\right)\cdot 163^{5} + \left(70 a^{2} + 59 a + 139\right)\cdot 163^{6} + \left(72 a^{2} + 51 a + 12\right)\cdot 163^{7} + \left(82 a^{2} + 126 a + 117\right)\cdot 163^{8} + \left(82 a^{2} + 94 a + 95\right)\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 56 a^{2} + 64 a + 16 + \left(74 a^{2} + 14 a + 109\right)\cdot 163 + \left(93 a^{2} + 162 a + 38\right)\cdot 163^{2} + \left(94 a^{2} + 42 a + 29\right)\cdot 163^{3} + \left(117 a^{2} + 54 a + 87\right)\cdot 163^{4} + \left(119 a^{2} + 10 a + 1\right)\cdot 163^{5} + \left(13 a^{2} + 2 a + 26\right)\cdot 163^{6} + \left(102 a^{2} + 52 a + 125\right)\cdot 163^{7} + \left(120 a^{2} + 68 a + 22\right)\cdot 163^{8} + \left(25 a^{2} + 44 a + 69\right)\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 97 a^{2} + 105 a + 153 + \left(83 a^{2} + 11 a + 97\right)\cdot 163 + \left(152 a^{2} + 138 a + 42\right)\cdot 163^{2} + \left(15 a^{2} + 153 a + 151\right)\cdot 163^{3} + \left(80 a^{2} + 102 a + 20\right)\cdot 163^{4} + \left(96 a + 97\right)\cdot 163^{5} + \left(125 a^{2} + 22 a + 110\right)\cdot 163^{6} + \left(43 a^{2} + 113 a + 124\right)\cdot 163^{7} + \left(154 a^{2} + 135 a + 16\right)\cdot 163^{8} + \left(37 a^{2} + 136 a + 126\right)\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 101 a^{2} + 73 a + 7 + \left(104 a^{2} + 7 a + 128\right)\cdot 163 + \left(137 a^{2} + 92 a + 1\right)\cdot 163^{2} + \left(139 a^{2} + 116 a + 72\right)\cdot 163^{3} + \left(158 a + 89\right)\cdot 163^{4} + \left(147 a^{2} + 19 a + 15\right)\cdot 163^{5} + \left(129 a^{2} + 108 a + 38\right)\cdot 163^{6} + \left(8 a^{2} + 96\right)\cdot 163^{7} + \left(156 a^{2} + 55 a + 134\right)\cdot 163^{8} + \left(102 a^{2} + 97 a + 136\right)\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 107 a^{2} + 83 a + 15 + \left(151 a^{2} + 95 a + 56\right)\cdot 163 + \left(128 a^{2} + 128 a + 118\right)\cdot 163^{2} + \left(132 a^{2} + 152 a + 36\right)\cdot 163^{3} + \left(112 a^{2} + 130 a + 33\right)\cdot 163^{4} + \left(85 a^{2} + 55 a + 106\right)\cdot 163^{5} + \left(92 a^{2} + 5 a + 5\right)\cdot 163^{6} + \left(130 a^{2} + 154 a + 36\right)\cdot 163^{7} + \left(95 a^{2} + 6 a + 5\right)\cdot 163^{8} + \left(154 a^{2} + 82 a + 138\right)\cdot 163^{9} +O(163^{10})\)
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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 |
| $1$ | $1$ | $()$ | $8$ |
| $9$ | $2$ | $(3,4)$ | $0$ |
| $18$ | $2$ | $(1,3)(4,5)(8,9)$ | $4$ |
| $27$ | $2$ | $(1,5)(3,4)$ | $0$ |
| $27$ | $2$ | $(1,5)(2,6)(3,4)$ | $0$ |
| $54$ | $2$ | $(1,5)(2,3)(4,6)(7,9)$ | $0$ |
| $6$ | $3$ | $(2,6,7)$ | $-4$ |
| $8$ | $3$ | $(1,5,8)(2,6,7)(3,4,9)$ | $-1$ |
| $12$ | $3$ | $(2,6,7)(3,4,9)$ | $2$ |
| $72$ | $3$ | $(1,2,3)(4,5,6)(7,9,8)$ | $2$ |
| $54$ | $4$ | $(1,4,5,3)(8,9)$ | $0$ |
| $162$ | $4$ | $(1,5)(2,3,6,4)(7,9)$ | $0$ |
| $36$ | $6$ | $(1,3)(2,6,7)(4,5)(8,9)$ | $-2$ |
| $36$ | $6$ | $(2,4,6,9,7,3)$ | $-2$ |
| $36$ | $6$ | $(2,6,7)(3,4)$ | $0$ |
| $36$ | $6$ | $(1,5,8)(2,6,7)(3,4)$ | $0$ |
| $54$ | $6$ | $(1,5)(2,7,6)(3,4)$ | $0$ |
| $72$ | $6$ | $(1,9,8,4,5,3)(2,6,7)$ | $1$ |
| $108$ | $6$ | $(1,5)(2,4,6,9,7,3)$ | $0$ |
| $216$ | $6$ | $(1,2,3,5,6,4)(7,9,8)$ | $0$ |
| $144$ | $9$ | $(1,2,4,5,6,9,8,7,3)$ | $-1$ |
| $108$ | $12$ | $(1,4,5,3)(2,6,7)(8,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.