Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 173 }$ to precision 13.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 173 }$: $ x^{3} + 2 x + 171 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 21 + 128\cdot 173 + 38\cdot 173^{2} + 29\cdot 173^{3} + 122\cdot 173^{4} + 145\cdot 173^{5} + 110\cdot 173^{6} + 148\cdot 173^{7} + 128\cdot 173^{8} + 22\cdot 173^{9} + 172\cdot 173^{10} + 30\cdot 173^{11} + 81\cdot 173^{12} +O\left(173^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 83 + 86\cdot 173 + 66\cdot 173^{2} + 130\cdot 173^{3} + 172\cdot 173^{4} + 26\cdot 173^{5} + 98\cdot 173^{6} + 124\cdot 173^{7} + 29\cdot 173^{8} + 136\cdot 173^{9} + 122\cdot 173^{10} + 73\cdot 173^{11} + 72\cdot 173^{12} +O\left(173^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 139 + 109\cdot 173 + 38\cdot 173^{2} + 41\cdot 173^{3} + 96\cdot 173^{4} + 4\cdot 173^{5} + 5\cdot 173^{6} + 119\cdot 173^{7} + 169\cdot 173^{8} + 53\cdot 173^{9} + 131\cdot 173^{10} + 6\cdot 173^{11} + 13\cdot 173^{12} +O\left(173^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 a^{2} + 15 a + 40 + \left(71 a^{2} + 48 a + 5\right)\cdot 173 + \left(83 a^{2} + 121 a + 167\right)\cdot 173^{2} + \left(10 a^{2} + 50 a + 25\right)\cdot 173^{3} + \left(161 a^{2} + 141 a + 8\right)\cdot 173^{4} + \left(134 a^{2} + 96 a + 172\right)\cdot 173^{5} + \left(126 a^{2} + 130 a + 62\right)\cdot 173^{6} + \left(142 a^{2} + 94 a + 2\right)\cdot 173^{7} + \left(49 a^{2} + 75 a + 61\right)\cdot 173^{8} + \left(56 a^{2} + 63 a + 159\right)\cdot 173^{9} + \left(154 a^{2} + 38 a + 97\right)\cdot 173^{10} + \left(143 a^{2} + 91 a + 29\right)\cdot 173^{11} + \left(61 a^{2} + 5 a + 95\right)\cdot 173^{12} +O\left(173^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 a^{2} + 38 a + 99 + \left(13 a^{2} + 113 a + 158\right)\cdot 173 + \left(85 a^{2} + 97 a + 53\right)\cdot 173^{2} + \left(145 a^{2} + 44 a + 148\right)\cdot 173^{3} + \left(74 a^{2} + 65 a + 123\right)\cdot 173^{4} + \left(42 a^{2} + 165 a + 48\right)\cdot 173^{5} + \left(57 a^{2} + 98 a + 143\right)\cdot 173^{6} + \left(168 a^{2} + 151 a + 151\right)\cdot 173^{7} + \left(164 a^{2} + 30 a + 156\right)\cdot 173^{8} + \left(162 a^{2} + 4 a + 70\right)\cdot 173^{9} + \left(7 a^{2} + 120 a + 133\right)\cdot 173^{10} + \left(2 a^{2} + 20 a + 128\right)\cdot 173^{11} + \left(2 a^{2} + 77 a + 130\right)\cdot 173^{12} +O\left(173^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 77 a^{2} + 63 a + 135 + \left(86 a^{2} + 68 a + 96\right)\cdot 173 + \left(106 a^{2} + 136 a + 153\right)\cdot 173^{2} + \left(75 a^{2} + 89 a + 21\right)\cdot 173^{3} + \left(98 a^{2} + 138 a + 92\right)\cdot 173^{4} + \left(83 a^{2} + 23 a + 2\right)\cdot 173^{5} + \left(164 a^{2} + 169 a + 81\right)\cdot 173^{6} + \left(38 a^{2} + 13 a + 51\right)\cdot 173^{7} + \left(71 a^{2} + 72 a + 106\right)\cdot 173^{8} + \left(154 a^{2} + 98 a + 50\right)\cdot 173^{9} + \left(2 a^{2} + 71 a + 27\right)\cdot 173^{10} + \left(126 a^{2} + 37 a + 120\right)\cdot 173^{11} + \left(26 a^{2} + 107 a + 82\right)\cdot 173^{12} +O\left(173^{ 13 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 116 a^{2} + 120 a + 42 + \left(88 a^{2} + 11 a + 86\right)\cdot 173 + \left(4 a^{2} + 127 a + 119\right)\cdot 173^{2} + \left(17 a^{2} + 77 a + 34\right)\cdot 173^{3} + \left(110 a^{2} + 139 a + 113\right)\cdot 173^{4} + \left(168 a^{2} + 83 a + 101\right)\cdot 173^{5} + \left(161 a^{2} + 116 a + 167\right)\cdot 173^{6} + \left(34 a^{2} + 99 a + 146\right)\cdot 173^{7} + \left(131 a^{2} + 66 a + 111\right)\cdot 173^{8} + \left(126 a^{2} + 105 a + 22\right)\cdot 173^{9} + \left(10 a^{2} + 14 a + 137\right)\cdot 173^{10} + \left(27 a^{2} + 61 a + 46\right)\cdot 173^{11} + \left(109 a^{2} + 90 a + 158\right)\cdot 173^{12} +O\left(173^{ 13 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 122 a^{2} + 157 a + 22 + \left(75 a^{2} + 98 a + 140\right)\cdot 173 + \left(119 a^{2} + 40 a + 170\right)\cdot 173^{2} + \left(32 a^{2} + 27 a + 79\right)\cdot 173^{3} + \left(146 a^{2} + 6 a + 40\right)\cdot 173^{4} + \left(88 a^{2} + 77 a + 67\right)\cdot 173^{5} + \left(153 a^{2} + 25 a + 66\right)\cdot 173^{6} + \left(136 a^{2} + 118 a + 124\right)\cdot 173^{7} + \left(94 a^{2} + 115 a + 137\right)\cdot 173^{8} + \left(71 a^{2} + 11 a + 170\right)\cdot 173^{9} + \left(32 a^{2} + 172 a + 8\right)\cdot 173^{10} + \left(58 a^{2} + 43 a + 145\right)\cdot 173^{11} + \left(169 a^{2} + 19 a + 99\right)\cdot 173^{12} +O\left(173^{ 13 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 147 a^{2} + 126 a + 113 + \left(10 a^{2} + 5 a + 53\right)\cdot 173 + \left(120 a^{2} + 169 a + 56\right)\cdot 173^{2} + \left(64 a^{2} + 55 a + 7\right)\cdot 173^{3} + \left(101 a^{2} + 28 a + 96\right)\cdot 173^{4} + \left(72 a + 122\right)\cdot 173^{5} + \left(28 a^{2} + 151 a + 129\right)\cdot 173^{6} + \left(170 a^{2} + 40 a + 168\right)\cdot 173^{7} + \left(6 a^{2} + 158 a + 135\right)\cdot 173^{8} + \left(120 a^{2} + 62 a + 4\right)\cdot 173^{9} + \left(137 a^{2} + 102 a + 34\right)\cdot 173^{10} + \left(161 a^{2} + 91 a + 110\right)\cdot 173^{11} + \left(149 a^{2} + 46 a + 131\right)\cdot 173^{12} +O\left(173^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(4,7,5)(6,9,8)$ |
| $(1,4,8)(2,5,6)(3,7,9)$ |
| $(1,3,2)(4,7,5)(6,8,9)$ |
| $(2,3)(4,6)(5,8)(7,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,9)(2,6)(3,8)(4,5)$ |
$0$ |
| $2$ |
$3$ |
$(1,3,2)(4,7,5)(6,8,9)$ |
$-3$ |
| $3$ |
$3$ |
$(1,3,2)(6,9,8)$ |
$0$ |
| $3$ |
$3$ |
$(1,2,3)(6,8,9)$ |
$0$ |
| $6$ |
$3$ |
$(1,4,8)(2,5,6)(3,7,9)$ |
$0$ |
| $6$ |
$3$ |
$(1,8,4)(2,6,5)(3,9,7)$ |
$0$ |
| $6$ |
$3$ |
$(1,5,8)(2,7,6)(3,4,9)$ |
$0$ |
| $9$ |
$6$ |
$(1,6,3,9,2,8)(4,5)$ |
$0$ |
| $9$ |
$6$ |
$(1,8,2,9,3,6)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
