Basic invariants
| Dimension: | $12$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(250275169231612352\)\(\medspace = 2^{6} \cdot 23^{5} \cdot 157^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.1.2811842368.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T315 |
| Parity: | odd |
| Projective image: | $S_3\wr S_3$ |
| Projective field: | Galois closure of 9.1.2811842368.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 173 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 173 }$:
\( x^{3} + 2x + 171 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 67 + 49\cdot 173 + 119\cdot 173^{2} + 167\cdot 173^{3} + 4\cdot 173^{4} + 54\cdot 173^{5} + 123\cdot 173^{6} + 171\cdot 173^{7} + 34\cdot 173^{8} + 3\cdot 173^{9} +O(173^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 134 + 166\cdot 173 + 20\cdot 173^{2} + 136\cdot 173^{3} + 93\cdot 173^{4} + 49\cdot 173^{5} + 131\cdot 173^{6} + 79\cdot 173^{7} + 123\cdot 173^{8} + 121\cdot 173^{9} +O(173^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 157 + 165\cdot 173 + 2\cdot 173^{2} + 118\cdot 173^{3} + 58\cdot 173^{4} + 140\cdot 173^{5} + 57\cdot 173^{6} + 169\cdot 173^{7} + 8\cdot 173^{8} + 147\cdot 173^{9} +O(173^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 31 a^{2} + 166 a + 135 + \left(158 a^{2} + 88 a + 9\right)\cdot 173 + \left(38 a^{2} + 126 a + 19\right)\cdot 173^{2} + \left(121 a^{2} + 93 a + 130\right)\cdot 173^{3} + \left(63 a^{2} + 64 a + 93\right)\cdot 173^{4} + \left(88 a^{2} + 74 a + 116\right)\cdot 173^{5} + \left(98 a^{2} + 55 a + 130\right)\cdot 173^{6} + \left(7 a^{2} + 4 a + 137\right)\cdot 173^{7} + \left(48 a^{2} + 78 a + 170\right)\cdot 173^{8} + \left(54 a^{2} + 96 a + 111\right)\cdot 173^{9} +O(173^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 84 a^{2} + 79 a + 131 + \left(32 a^{2} + 98 a + 1\right)\cdot 173 + \left(5 a^{2} + 20 a + 165\right)\cdot 173^{2} + \left(117 a^{2} + 121 a + 46\right)\cdot 173^{3} + \left(61 a^{2} + 122 a + 136\right)\cdot 173^{4} + \left(155 a^{2} + 155 a + 11\right)\cdot 173^{5} + \left(134 a^{2} + 77 a + 134\right)\cdot 173^{6} + \left(42 a^{2} + 157 a + 19\right)\cdot 173^{7} + \left(58 a^{2} + 148 a + 88\right)\cdot 173^{8} + \left(137 a^{2} + 139 a + 110\right)\cdot 173^{9} +O(173^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 101 a^{2} + 59 a + 96 + \left(168 a^{2} + 5 a + 125\right)\cdot 173 + \left(152 a^{2} + 163 a + 73\right)\cdot 173^{2} + \left(95 a^{2} + 16 a + 76\right)\cdot 173^{3} + \left(112 a^{2} + 132 a + 146\right)\cdot 173^{4} + \left(4 a^{2} + 71 a + 156\right)\cdot 173^{5} + \left(109 a^{2} + 86 a + 41\right)\cdot 173^{6} + \left(90 a^{2} + 44 a + 141\right)\cdot 173^{7} + \left(108 a^{2} + 172 a + 39\right)\cdot 173^{8} + \left(149 a^{2} + 111 a + 69\right)\cdot 173^{9} +O(173^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 153 a^{2} + 124 a + 67 + \left(95 a^{2} + 106 a + 157\right)\cdot 173 + \left(150 a^{2} + 126 a + 167\right)\cdot 173^{2} + \left(75 a^{2} + 150 a + 11\right)\cdot 173^{3} + \left(10 a^{2} + 17 a + 138\right)\cdot 173^{4} + \left(148 a^{2} + 9 a + 80\right)\cdot 173^{5} + \left(12 a^{2} + 129 a + 16\right)\cdot 173^{6} + \left(158 a^{2} + 3 a + 50\right)\cdot 173^{7} + \left(158 a^{2} + 143 a + 30\right)\cdot 173^{8} + \left(71 a^{2} + 164 a + 20\right)\cdot 173^{9} +O(173^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 161 a^{2} + 35 a + 3 + \left(144 a^{2} + 69 a + 94\right)\cdot 173 + \left(14 a^{2} + 162 a + 62\right)\cdot 173^{2} + \left(133 a^{2} + 34 a + 68\right)\cdot 173^{3} + \left(171 a^{2} + 91 a + 52\right)\cdot 173^{4} + \left(12 a^{2} + 118 a + 110\right)\cdot 173^{5} + \left(102 a^{2} + 8 a + 32\right)\cdot 173^{6} + \left(39 a^{2} + 144 a + 73\right)\cdot 173^{7} + \left(6 a^{2} + 24 a + 76\right)\cdot 173^{8} + \left(59 a^{2} + 94 a + 121\right)\cdot 173^{9} +O(173^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 162 a^{2} + 56 a + 79 + \left(91 a^{2} + 150 a + 94\right)\cdot 173 + \left(156 a^{2} + 92 a + 60\right)\cdot 173^{2} + \left(148 a^{2} + 101 a + 109\right)\cdot 173^{3} + \left(98 a^{2} + 90 a + 140\right)\cdot 173^{4} + \left(109 a^{2} + 89 a + 144\right)\cdot 173^{5} + \left(61 a^{2} + 161 a + 23\right)\cdot 173^{6} + \left(7 a^{2} + 164 a + 22\right)\cdot 173^{7} + \left(139 a^{2} + 124 a + 119\right)\cdot 173^{8} + \left(46 a^{2} + 84 a + 159\right)\cdot 173^{9} +O(173^{10})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $12$ |
| $9$ | $2$ | $(4,7)$ | $4$ |
| $18$ | $2$ | $(1,4)(2,7)(3,9)$ | $2$ |
| $27$ | $2$ | $(1,2)(4,7)(5,6)$ | $0$ |
| $27$ | $2$ | $(4,7)(5,6)$ | $0$ |
| $54$ | $2$ | $(1,2)(4,5)(6,7)(8,9)$ | $2$ |
| $6$ | $3$ | $(5,6,8)$ | $0$ |
| $8$ | $3$ | $(1,3,2)(4,9,7)(5,8,6)$ | $3$ |
| $12$ | $3$ | $(4,9,7)(5,8,6)$ | $-3$ |
| $72$ | $3$ | $(1,5,4)(2,6,7)(3,8,9)$ | $0$ |
| $54$ | $4$ | $(4,5,7,6)(8,9)$ | $0$ |
| $162$ | $4$ | $(1,2)(4,5,7,6)(8,9)$ | $0$ |
| $36$ | $6$ | $(1,4)(2,7)(3,9)(5,6,8)$ | $2$ |
| $36$ | $6$ | $(4,5,9,8,7,6)$ | $-1$ |
| $36$ | $6$ | $(4,7)(5,6,8)$ | $-2$ |
| $36$ | $6$ | $(1,2,3)(4,7)(5,6,8)$ | $1$ |
| $54$ | $6$ | $(1,2)(4,7)(5,8,6)$ | $0$ |
| $72$ | $6$ | $(1,9,3,7,2,4)(5,6,8)$ | $-1$ |
| $108$ | $6$ | $(1,2)(4,5,9,8,7,6)$ | $-1$ |
| $216$ | $6$ | $(1,5,7,2,6,4)(3,8,9)$ | $0$ |
| $144$ | $9$ | $(1,5,9,3,8,7,2,6,4)$ | $0$ |
| $108$ | $12$ | $(1,7,2,4)(3,9)(5,6,8)$ | $0$ |