Basic invariants
| Dimension: | $12$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(49899466530704241\)\(\medspace = 3^{5} \cdot 11^{4} \cdot 107^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.5.38930641497.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T315 |
| Parity: | even |
| Determinant: | 1.321.2t1.a.a |
| Projective image: | $S_3\wr S_3$ |
| Projective stem field: | Galois closure of 9.5.38930641497.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{8} - 2x^{7} + 9x^{6} + 2x^{5} - 8x^{4} - x^{3} - 2x^{2} + 4x - 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 19 a^{2} + 13 a + 103 + \left(10 a^{2} + 34 a + 57\right)\cdot 167 + \left(75 a^{2} + 118 a + 127\right)\cdot 167^{2} + \left(98 a^{2} + 155 a + 125\right)\cdot 167^{3} + \left(6 a^{2} + 157 a + 111\right)\cdot 167^{4} + \left(16 a^{2} + 116 a + 121\right)\cdot 167^{5} + \left(57 a^{2} + 16 a + 51\right)\cdot 167^{6} + \left(10 a^{2} + 40 a + 1\right)\cdot 167^{7} + \left(35 a^{2} + 68 a + 71\right)\cdot 167^{8} + \left(41 a^{2} + 87 a + 43\right)\cdot 167^{9} +O(167^{10})\)
| $r_{ 2 }$ |
$=$ |
\( 21 a^{2} + 125 a + 87 + \left(103 a^{2} + 135 a + 38\right)\cdot 167 + \left(8 a^{2} + 161 a + 89\right)\cdot 167^{2} + \left(102 a^{2} + 93 a + 120\right)\cdot 167^{3} + \left(13 a^{2} + 126 a + 120\right)\cdot 167^{4} + \left(138 a^{2} + 67 a + 95\right)\cdot 167^{5} + \left(111 a^{2} + 34 a + 33\right)\cdot 167^{6} + \left(17 a^{2} + 146 a + 22\right)\cdot 167^{7} + \left(45 a^{2} + 162 a + 159\right)\cdot 167^{8} + \left(80 a^{2} + 62 a + 32\right)\cdot 167^{9} +O(167^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 37 a^{2} + 31 a + 106 + \left(130 a^{2} + 75 a + 53\right)\cdot 167 + \left(83 a^{2} + 111 a + 50\right)\cdot 167^{2} + \left(101 a^{2} + 130 a + 62\right)\cdot 167^{3} + \left(62 a^{2} + 33 a + 15\right)\cdot 167^{4} + \left(98 a^{2} + 25 a + 133\right)\cdot 167^{5} + \left(134 a^{2} + 82 a + 139\right)\cdot 167^{6} + \left(54 a^{2} + 107 a + 139\right)\cdot 167^{7} + \left(89 a^{2} + 59 a + 142\right)\cdot 167^{8} + \left(35 a^{2} + 28 a + 46\right)\cdot 167^{9} +O(167^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 62 a^{2} + 45 a + 120 + \left(16 a^{2} + 13 a + 7\right)\cdot 167 + \left(48 a^{2} + 119 a + 120\right)\cdot 167^{2} + \left(40 a^{2} + 156 a + 153\right)\cdot 167^{3} + \left(165 a^{2} + 23 a + 131\right)\cdot 167^{4} + \left(88 a^{2} + 31 a + 137\right)\cdot 167^{5} + \left(20 a^{2} + 69 a + 130\right)\cdot 167^{6} + \left(47 a^{2} + 154 a + 48\right)\cdot 167^{7} + \left(39 a^{2} + 102 a + 160\right)\cdot 167^{8} + \left(101 a^{2} + 120 a + 127\right)\cdot 167^{9} +O(167^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 109 a^{2} + 11 a + 108 + \left(100 a^{2} + 123 a + 82\right)\cdot 167 + \left(74 a^{2} + 60 a + 7\right)\cdot 167^{2} + \left(130 a^{2} + 109 a + 86\right)\cdot 167^{3} + \left(90 a^{2} + 6 a + 35\right)\cdot 167^{4} + \left(97 a^{2} + 74 a + 129\right)\cdot 167^{5} + \left(87 a^{2} + 50 a + 31\right)\cdot 167^{6} + \left(94 a^{2} + 80 a + 158\right)\cdot 167^{7} + \left(32 a^{2} + 111 a + 44\right)\cdot 167^{8} + \left(51 a^{2} + 75 a + 64\right)\cdot 167^{9} +O(167^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 118 a^{2} + 46 a + 103 + \left(67 a^{2} + 165 a + 24\right)\cdot 167 + \left(145 a^{2} + 37 a + 73\right)\cdot 167^{2} + \left(164 a^{2} + 123 a + 11\right)\cdot 167^{3} + \left(83 a^{2} + 128 a + 31\right)\cdot 167^{4} + \left(73 a^{2} + 112 a + 121\right)\cdot 167^{5} + \left(135 a^{2} + 42 a + 54\right)\cdot 167^{6} + \left(164 a^{2} + 150 a + 41\right)\cdot 167^{7} + \left(45 a^{2} + 149 a + 80\right)\cdot 167^{8} + \left(139 a^{2} + 85 a + 138\right)\cdot 167^{9} +O(167^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 154 a^{2} + 76 a + 104 + \left(82 a^{2} + 155 a + 95\right)\cdot 167 + \left(140 a^{2} + 9 a + 161\right)\cdot 167^{2} + \left(128 a^{2} + 54 a + 65\right)\cdot 167^{3} + \left(84 a^{2} + 14 a + 90\right)\cdot 167^{4} + \left(4 a^{2} + 23 a + 77\right)\cdot 167^{5} + \left(11 a^{2} + 55 a + 86\right)\cdot 167^{6} + \left(122 a^{2} + 29 a + 64\right)\cdot 167^{7} + \left(81 a^{2} + 81 a + 24\right)\cdot 167^{8} + \left(93 a^{2} + 127 a + 36\right)\cdot 167^{9} +O(167^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 157 a^{2} + 165 a + 79 + \left(124 a^{2} + 47 a + 92\right)\cdot 167 + \left(111 a^{2} + 84 a + 131\right)\cdot 167^{2} + \left(136 a^{2} + 121 a + 25\right)\cdot 167^{3} + \left(49 a^{2} + 130 a + 35\right)\cdot 167^{4} + \left(69 a^{2} + 64 a + 36\right)\cdot 167^{5} + \left(69 a^{2} + 131 a + 109\right)\cdot 167^{6} + \left(97 a^{2} + 89 a + 73\right)\cdot 167^{7} + \left(40 a^{2} + 41 a + 152\right)\cdot 167^{8} + \left(111 a^{2} + 137 a + 91\right)\cdot 167^{9} +O(167^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 158 a^{2} + 156 a + 28 + \left(31 a^{2} + 84 a + 48\right)\cdot 167 + \left(147 a^{2} + 131 a + 74\right)\cdot 167^{2} + \left(98 a^{2} + 56 a + 16\right)\cdot 167^{3} + \left(110 a^{2} + 45 a + 96\right)\cdot 167^{4} + \left(81 a^{2} + 152 a + 149\right)\cdot 167^{5} + \left(40 a^{2} + 18 a + 29\right)\cdot 167^{6} + \left(59 a^{2} + 37 a + 118\right)\cdot 167^{7} + \left(91 a^{2} + 57 a + 166\right)\cdot 167^{8} + \left(14 a^{2} + 109 a + 85\right)\cdot 167^{9} +O(167^{10})\)
| |
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 |
| $1$ | $1$ | $()$ | $12$ |
| $9$ | $2$ | $(1,3)$ | $4$ |
| $18$ | $2$ | $(1,5)(3,6)(7,8)$ | $2$ |
| $27$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $27$ | $2$ | $(1,3)(5,6)$ | $0$ |
| $54$ | $2$ | $(1,3)(2,5)(4,6)(8,9)$ | $2$ |
| $6$ | $3$ | $(2,4,9)$ | $0$ |
| $8$ | $3$ | $(1,3,7)(2,4,9)(5,6,8)$ | $3$ |
| $12$ | $3$ | $(2,4,9)(5,6,8)$ | $-3$ |
| $72$ | $3$ | $(1,5,2)(3,6,4)(7,8,9)$ | $0$ |
| $54$ | $4$ | $(1,6,3,5)(7,8)$ | $0$ |
| $162$ | $4$ | $(1,4,3,2)(6,8)(7,9)$ | $0$ |
| $36$ | $6$ | $(1,5)(2,4,9)(3,6)(7,8)$ | $2$ |
| $36$ | $6$ | $(1,2,3,4,7,9)$ | $-1$ |
| $36$ | $6$ | $(1,3)(2,4,9)$ | $-2$ |
| $36$ | $6$ | $(1,3)(2,4,9)(5,6,8)$ | $1$ |
| $54$ | $6$ | $(1,3)(2,9,4)(5,6)$ | $0$ |
| $72$ | $6$ | $(1,5,3,6,7,8)(2,4,9)$ | $-1$ |
| $108$ | $6$ | $(1,3)(2,6,4,8,9,5)$ | $-1$ |
| $216$ | $6$ | $(1,6,4,3,5,2)(7,8,9)$ | $0$ |
| $144$ | $9$ | $(1,5,2,3,6,4,7,8,9)$ | $0$ |
| $108$ | $12$ | $(1,6,3,5)(2,4,9)(7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.