Basic invariants
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} + 2 x + 171 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 65 + 153\cdot 173 + 141\cdot 173^{2} + 4\cdot 173^{3} + 35\cdot 173^{4} + 163\cdot 173^{5} + 80\cdot 173^{6} + 146\cdot 173^{7} + 103\cdot 173^{8} + 91\cdot 173^{9} +O\left(173^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 78 + 120\cdot 173 + 54\cdot 173^{2} + 90\cdot 173^{3} + 30\cdot 173^{4} + 155\cdot 173^{5} + 123\cdot 173^{6} + 66\cdot 173^{7} + 96\cdot 173^{8} + 137\cdot 173^{9} +O\left(173^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 151 + 172\cdot 173 + 97\cdot 173^{2} + 112\cdot 173^{3} + 69\cdot 173^{4} + 64\cdot 173^{5} + 145\cdot 173^{6} + 145\cdot 173^{7} + 120\cdot 173^{8} + 62\cdot 173^{9} +O\left(173^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 42 a^{2} + 148 a + 124 + \left(69 a^{2} + 28 a + 163\right)\cdot 173 + \left(27 a^{2} + 136 a + 147\right)\cdot 173^{2} + \left(60 a^{2} + 30 a + 54\right)\cdot 173^{3} + \left(129 a^{2} + 75 a + 124\right)\cdot 173^{4} + \left(99 a^{2} + 157 a + 136\right)\cdot 173^{5} + \left(120 a^{2} + 62 a + 9\right)\cdot 173^{6} + \left(133 a^{2} + 79 a + 66\right)\cdot 173^{7} + \left(124 a^{2} + 31 a + 76\right)\cdot 173^{8} + \left(65 a^{2} + 41 a + 54\right)\cdot 173^{9} +O\left(173^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 72 a^{2} + 89 a + 161 + \left(76 a^{2} + 163 a + 169\right)\cdot 173 + \left(170 a^{2} + 162 a + 132\right)\cdot 173^{2} + \left(92 a^{2} + 21 a + 137\right)\cdot 173^{3} + \left(63 a^{2} + 89 a + 87\right)\cdot 173^{4} + \left(118 a^{2} + 129 a + 26\right)\cdot 173^{5} + \left(71 a^{2} + 146 a + 72\right)\cdot 173^{6} + \left(108 a^{2} + 169 a + 79\right)\cdot 173^{7} + \left(16 a^{2} + 53 a + 120\right)\cdot 173^{8} + \left(34 a^{2} + 13 a + 96\right)\cdot 173^{9} +O\left(173^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 116 a^{2} + 147 a + 162 + \left(72 a^{2} + 101 a + 164\right)\cdot 173 + \left(158 a^{2} + a + 116\right)\cdot 173^{2} + \left(136 a^{2} + 5 a + 138\right)\cdot 173^{3} + \left(34 a^{2} + 162 a + 164\right)\cdot 173^{4} + \left(61 a^{2} + 131 a + 7\right)\cdot 173^{5} + \left(147 a^{2} + 118 a\right)\cdot 173^{6} + \left(117 a^{2} + 7 a + 92\right)\cdot 173^{7} + \left(154 a^{2} + 69 a + 131\right)\cdot 173^{8} + \left(31 a^{2} + a + 93\right)\cdot 173^{9} +O\left(173^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 140 a^{2} + 23 a + 24 + \left(39 a^{2} + 38 a + 9\right)\cdot 173 + \left(88 a^{2} + 5 a + 56\right)\cdot 173^{2} + \left(11 a^{2} + 136 a + 105\right)\cdot 173^{3} + \left(54 a^{2} + 142 a + 81\right)\cdot 173^{4} + \left(46 a^{2} + 139 a + 65\right)\cdot 173^{5} + \left(46 a^{2} + 128 a + 141\right)\cdot 173^{6} + \left(62 a^{2} + 156 a + 143\right)\cdot 173^{7} + \left(20 a^{2} + 41 a + 167\right)\cdot 173^{8} + \left(112 a^{2} + 32 a\right)\cdot 173^{9} +O\left(173^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 158 a^{2} + 110 a + 45 + \left(23 a^{2} + 80 a + 42\right)\cdot 173 + \left(17 a^{2} + 8 a + 159\right)\cdot 173^{2} + \left(116 a^{2} + 146 a + 110\right)\cdot 173^{3} + \left(74 a^{2} + 94 a + 102\right)\cdot 173^{4} + \left(166 a^{2} + 84 a + 90\right)\cdot 173^{5} + \left(126 a^{2} + 80 a + 30\right)\cdot 173^{6} + \left(119 a^{2} + 168 a + 152\right)\cdot 173^{7} + \left(a^{2} + 49 a + 42\right)\cdot 173^{8} + \left(107 a^{2} + 158 a + 136\right)\cdot 173^{9} +O\left(173^{ 10 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 164 a^{2} + 2 a + 56 + \left(63 a^{2} + 106 a + 41\right)\cdot 173 + \left(57 a^{2} + 31 a + 130\right)\cdot 173^{2} + \left(101 a^{2} + 6 a + 109\right)\cdot 173^{3} + \left(162 a^{2} + 128 a + 168\right)\cdot 173^{4} + \left(26 a^{2} + 48 a + 154\right)\cdot 173^{5} + \left(6 a^{2} + 154 a + 87\right)\cdot 173^{6} + \left(150 a^{2} + 109 a + 145\right)\cdot 173^{7} + \left(27 a^{2} + 99 a + 4\right)\cdot 173^{8} + \left(168 a^{2} + 99 a + 18\right)\cdot 173^{9} +O\left(173^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,5,4)(2,6,7)(3,8,9)$ |
| $(1,2)(4,7)(5,6)$ |
| $(4,5)(6,7)(8,9)$ |
| $(1,2,3)(4,7,9)(5,6,8)$ |
| $(4,7,9)(5,8,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,2)(4,7)(5,6)$ | $0$ |
| $9$ | $2$ | $(4,5)(6,7)(8,9)$ | $-2$ |
| $9$ | $2$ | $(1,2)(4,6)(5,7)(8,9)$ | $0$ |
| $2$ | $3$ | $(1,2,3)(4,7,9)(5,6,8)$ | $-3$ |
| $6$ | $3$ | $(1,5,4)(2,6,7)(3,8,9)$ | $0$ |
| $6$ | $3$ | $(1,2,3)(5,8,6)$ | $0$ |
| $12$ | $3$ | $(1,8,4)(2,5,7)(3,6,9)$ | $0$ |
| $18$ | $6$ | $(1,6,4,2,5,7)(3,8,9)$ | $0$ |
| $18$ | $6$ | $(1,2,3)(4,6,9,5,7,8)$ | $1$ |
| $18$ | $6$ | $(1,5,2,8,3,6)(4,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
