Basic invariants
| Dimension: | $12$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(63291734159899019\)\(\medspace = 59^{5} \cdot 97^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.1175384017.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T315 |
| Parity: | odd |
| Determinant: | 1.59.2t1.a.a |
| Projective image: | $S_3\wr S_3$ |
| Projective stem field: | Galois closure of 9.1.1175384017.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} + x^{7} - x^{6} + x^{5} + x^{4} + 2x^{3} + x^{2} - x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 167 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$:
\( x^{3} + 7x + 162 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 26 a^{2} + 28 a + 24 + \left(91 a^{2} + 17 a + 41\right)\cdot 167 + \left(15 a^{2} + 95 a + 67\right)\cdot 167^{2} + \left(157 a^{2} + 133 a + 19\right)\cdot 167^{3} + \left(11 a^{2} + 106 a + 42\right)\cdot 167^{4} + \left(164 a^{2} + 97 a + 123\right)\cdot 167^{5} + \left(63 a^{2} + 14 a + 156\right)\cdot 167^{6} + \left(114 a^{2} + 22 a + 23\right)\cdot 167^{7} + \left(15 a^{2} + 38 a + 113\right)\cdot 167^{8} + \left(77 a^{2} + 133 a + 156\right)\cdot 167^{9} +O(167^{10})\)
| $r_{ 2 }$ |
$=$ |
\( 29 a^{2} + 128 a + 67 + \left(29 a^{2} + 44 a + 55\right)\cdot 167 + \left(93 a^{2} + 84 a + 95\right)\cdot 167^{2} + \left(133 a^{2} + 92 a + 134\right)\cdot 167^{3} + \left(91 a^{2} + 148 a + 16\right)\cdot 167^{4} + \left(21 a^{2} + 161 a + 39\right)\cdot 167^{5} + \left(a^{2} + 6 a + 133\right)\cdot 167^{6} + \left(10 a^{2} + 4 a + 84\right)\cdot 167^{7} + \left(58 a^{2} + 82 a + 18\right)\cdot 167^{8} + \left(165 a^{2} + 116 a + 3\right)\cdot 167^{9} +O(167^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 31 a^{2} + 118 a + 132 + \left(19 a^{2} + 17 a + 8\right)\cdot 167 + \left(119 a^{2} + 99 a + 105\right)\cdot 167^{2} + \left(98 a^{2} + 66 a + 27\right)\cdot 167^{3} + \left(48 a^{2} + 6 a + 149\right)\cdot 167^{4} + \left(18 a^{2} + 30 a + 23\right)\cdot 167^{5} + \left(75 a^{2} + 14 a + 33\right)\cdot 167^{6} + \left(20 a^{2} + 28 a + 22\right)\cdot 167^{7} + \left(62 a^{2} + 78 a + 93\right)\cdot 167^{8} + \left(123 a^{2} + 55 a + 85\right)\cdot 167^{9} +O(167^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 47 a^{2} + 157 a + 122 + \left(61 a^{2} + 141 a + 68\right)\cdot 167 + \left(112 a^{2} + 56 a + 129\right)\cdot 167^{2} + \left(123 a^{2} + 166 a + 141\right)\cdot 167^{3} + \left(35 a^{2} + 88 a + 41\right)\cdot 167^{4} + \left(156 a^{2} + 117 a + 142\right)\cdot 167^{5} + \left(70 a^{2} + 148 a + 77\right)\cdot 167^{6} + \left(44 a^{2} + 54 a + 31\right)\cdot 167^{7} + \left(4 a^{2} + 77 a + 4\right)\cdot 167^{8} + \left(49 a^{2} + 43 a + 137\right)\cdot 167^{9} +O(167^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 73 a^{2} + 121 a + 61 + \left(122 a^{2} + 52 a + 34\right)\cdot 167 + \left(72 a^{2} + 127 a + 16\right)\cdot 167^{2} + \left(151 a^{2} + 71 a + 72\right)\cdot 167^{3} + \left(49 a^{2} + 133 a + 101\right)\cdot 167^{4} + \left(119 a^{2} + 104 a + 91\right)\cdot 167^{5} + \left(11 a^{2} + 163 a + 68\right)\cdot 167^{6} + \left(91 a^{2} + 34 a + 117\right)\cdot 167^{7} + \left(92 a^{2} + 38 a + 87\right)\cdot 167^{8} + \left(110 a^{2} + 20 a + 151\right)\cdot 167^{9} +O(167^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 94 a^{2} + 149 a + 63 + \left(14 a^{2} + 7 a + 73\right)\cdot 167 + \left(39 a^{2} + 15 a + 121\right)\cdot 167^{2} + \left(53 a^{2} + 34 a + 35\right)\cdot 167^{3} + \left(119 a^{2} + 138 a + 42\right)\cdot 167^{4} + \left(13 a^{2} + 118 a + 145\right)\cdot 167^{5} + \left(32 a^{2} + 3 a + 7\right)\cdot 167^{6} + \left(8 a^{2} + 90 a + 85\right)\cdot 167^{7} + \left(147 a^{2} + 51 a + 2\right)\cdot 167^{8} + \left(40 a^{2} + 157 a + 99\right)\cdot 167^{9} +O(167^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 107 a^{2} + 88 a + 97 + \left(118 a^{2} + 104 a + 27\right)\cdot 167 + \left(121 a^{2} + 150 a + 117\right)\cdot 167^{2} + \left(101 a^{2} + 7 a + 41\right)\cdot 167^{3} + \left(26 a^{2} + 12 a + 102\right)\cdot 167^{4} + \left(127 a^{2} + 142 a + 86\right)\cdot 167^{5} + \left(90 a^{2} + 145 a + 50\right)\cdot 167^{6} + \left(136 a^{2} + 134 a + 7\right)\cdot 167^{7} + \left(46 a^{2} + 6 a + 133\right)\cdot 167^{8} + \left(45 a^{2} + 162 a + 110\right)\cdot 167^{9} +O(167^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 113 a^{2} + 118 a + 25 + \left(108 a^{2} + 81 a + 137\right)\cdot 167 + \left(80 a^{2} + 75 a + 108\right)\cdot 167^{2} + \left(34 a^{2} + 20 a + 138\right)\cdot 167^{3} + \left(22 a^{2} + 134 a + 27\right)\cdot 167^{4} + \left(17 a + 92\right)\cdot 167^{5} + \left(43 a^{2} + 160 a + 47\right)\cdot 167^{6} + \left(106 a^{2} + 102 a + 21\right)\cdot 167^{7} + \left(55 a^{2} + 139 a + 138\right)\cdot 167^{8} + \left(27 a^{2} + 12 a + 152\right)\cdot 167^{9} +O(167^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 148 a^{2} + 95 a + 77 + \left(102 a^{2} + 32 a + 54\right)\cdot 167 + \left(13 a^{2} + 131 a + 74\right)\cdot 167^{2} + \left(148 a^{2} + 74 a + 56\right)\cdot 167^{3} + \left(94 a^{2} + 66 a + 144\right)\cdot 167^{4} + \left(47 a^{2} + 44 a + 90\right)\cdot 167^{5} + \left(112 a^{2} + 10 a + 92\right)\cdot 167^{6} + \left(136 a^{2} + 29 a + 107\right)\cdot 167^{7} + \left(18 a^{2} + 156 a + 77\right)\cdot 167^{8} + \left(29 a^{2} + 133 a + 105\right)\cdot 167^{9} +O(167^{10})\)
| |
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 |
| $1$ | $1$ | $()$ | $12$ |
| $9$ | $2$ | $(2,4)$ | $4$ |
| $18$ | $2$ | $(1,2)(4,7)(5,8)$ | $2$ |
| $27$ | $2$ | $(1,7)(2,4)(3,6)$ | $0$ |
| $27$ | $2$ | $(2,4)(3,6)$ | $0$ |
| $54$ | $2$ | $(1,7)(2,3)(4,6)(5,9)$ | $2$ |
| $6$ | $3$ | $(3,6,9)$ | $0$ |
| $8$ | $3$ | $(1,8,7)(2,5,4)(3,9,6)$ | $3$ |
| $12$ | $3$ | $(2,5,4)(3,9,6)$ | $-3$ |
| $72$ | $3$ | $(1,3,2)(4,7,6)(5,8,9)$ | $0$ |
| $54$ | $4$ | $(2,3,4,6)(5,9)$ | $0$ |
| $162$ | $4$ | $(1,7)(2,3,4,6)(5,9)$ | $0$ |
| $36$ | $6$ | $(1,2)(3,6,9)(4,7)(5,8)$ | $2$ |
| $36$ | $6$ | $(2,3,5,9,4,6)$ | $-1$ |
| $36$ | $6$ | $(2,4)(3,6,9)$ | $-2$ |
| $36$ | $6$ | $(1,7,8)(2,4)(3,6,9)$ | $1$ |
| $54$ | $6$ | $(1,7)(2,4)(3,9,6)$ | $0$ |
| $72$ | $6$ | $(1,5,8,4,7,2)(3,6,9)$ | $-1$ |
| $108$ | $6$ | $(1,7)(2,3,5,9,4,6)$ | $-1$ |
| $216$ | $6$ | $(1,3,4,7,6,2)(5,8,9)$ | $0$ |
| $144$ | $9$ | $(1,3,5,8,9,4,7,6,2)$ | $0$ |
| $108$ | $12$ | $(1,4,7,2)(3,6,9)(5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.