Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 173 }$ to precision 20.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 173 }$: $ x^{3} + 2 x + 171 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 37 + 52\cdot 173 + 98\cdot 173^{2} + 162\cdot 173^{3} + 116\cdot 173^{4} + 158\cdot 173^{5} + 50\cdot 173^{6} + 93\cdot 173^{7} + 151\cdot 173^{8} + 168\cdot 173^{9} + 60\cdot 173^{10} + 67\cdot 173^{11} + 49\cdot 173^{12} + 44\cdot 173^{13} + 98\cdot 173^{14} + 96\cdot 173^{15} + 167\cdot 173^{16} + 55\cdot 173^{17} + 89\cdot 173^{18} + 49\cdot 173^{19} +O\left(173^{ 20 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 101 + 152\cdot 173 + 88\cdot 173^{2} + 166\cdot 173^{3} + 101\cdot 173^{4} + 63\cdot 173^{5} + 96\cdot 173^{6} + 106\cdot 173^{7} + 85\cdot 173^{8} + 41\cdot 173^{9} + 14\cdot 173^{10} + 82\cdot 173^{11} + 83\cdot 173^{12} + 122\cdot 173^{13} + 34\cdot 173^{14} + 39\cdot 173^{15} + 29\cdot 173^{16} + 143\cdot 173^{17} + 13\cdot 173^{18} + 148\cdot 173^{19} +O\left(173^{ 20 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 114 + 77\cdot 173 + 95\cdot 173^{2} + 152\cdot 173^{3} + 79\cdot 173^{4} + 164\cdot 173^{5} + 129\cdot 173^{6} + 29\cdot 173^{7} + 63\cdot 173^{8} + 46\cdot 173^{9} + 34\cdot 173^{10} + 136\cdot 173^{11} + 103\cdot 173^{12} + 66\cdot 173^{13} + 4\cdot 173^{14} + 145\cdot 173^{15} + 159\cdot 173^{16} + 154\cdot 173^{17} + 155\cdot 173^{18} + 85\cdot 173^{19} +O\left(173^{ 20 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 a^{2} + 98 a + 27 + \left(16 a + 44\right)\cdot 173 + \left(35 a^{2} + 168 a + 121\right)\cdot 173^{2} + \left(168 a^{2} + 33 a + 108\right)\cdot 173^{3} + \left(93 a^{2} + 4 a + 15\right)\cdot 173^{4} + \left(65 a^{2} + 126 a + 74\right)\cdot 173^{5} + \left(86 a^{2} + 116 a + 68\right)\cdot 173^{6} + \left(86 a^{2} + 156\right)\cdot 173^{7} + \left(7 a^{2} + 156 a + 142\right)\cdot 173^{8} + \left(159 a^{2} + 106 a + 167\right)\cdot 173^{9} + \left(167 a^{2} + 78 a + 152\right)\cdot 173^{10} + \left(127 a^{2} + 72 a + 133\right)\cdot 173^{11} + \left(63 a^{2} + 121 a + 167\right)\cdot 173^{12} + \left(102 a^{2} + 127 a + 106\right)\cdot 173^{13} + \left(169 a^{2} + 137 a + 81\right)\cdot 173^{14} + \left(57 a^{2} + 86 a + 10\right)\cdot 173^{15} + \left(81 a^{2} + 60 a + 148\right)\cdot 173^{16} + \left(133 a^{2} + 143 a + 148\right)\cdot 173^{17} + \left(28 a^{2} + 9 a + 50\right)\cdot 173^{18} + \left(97 a^{2} + 88 a + 61\right)\cdot 173^{19} +O\left(173^{ 20 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 51 a^{2} + 30 a + 106 + \left(77 a^{2} + 13 a + 31\right)\cdot 173 + \left(134 a^{2} + 163 a + 23\right)\cdot 173^{2} + \left(39 a^{2} + 50 a + 168\right)\cdot 173^{3} + \left(93 a^{2} + 2 a + 129\right)\cdot 173^{4} + \left(100 a^{2} + 46 a + 120\right)\cdot 173^{5} + \left(10 a^{2} + 129 a + 82\right)\cdot 173^{6} + \left(159 a^{2} + 78 a + 22\right)\cdot 173^{7} + \left(108 a^{2} + 4 a + 105\right)\cdot 173^{8} + \left(155 a^{2} + 107 a + 105\right)\cdot 173^{9} + \left(63 a^{2} + 156 a + 129\right)\cdot 173^{10} + \left(138 a^{2} + 123 a + 147\right)\cdot 173^{11} + \left(13 a^{2} + 128 a + 158\right)\cdot 173^{12} + \left(37 a^{2} + 115 a + 19\right)\cdot 173^{13} + \left(26 a^{2} + 85 a + 121\right)\cdot 173^{14} + \left(158 a^{2} + 73 a + 28\right)\cdot 173^{15} + \left(4 a^{2} + 74 a + 46\right)\cdot 173^{16} + \left(47 a^{2} + 18 a + 91\right)\cdot 173^{17} + \left(103 a^{2} + 20 a + 92\right)\cdot 173^{18} + \left(36 a^{2} + 45 a + 153\right)\cdot 173^{19} +O\left(173^{ 20 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 87 a^{2} + 45 a + 154 + \left(95 a^{2} + 143 a + 55\right)\cdot 173 + \left(3 a^{2} + 14 a + 79\right)\cdot 173^{2} + \left(138 a^{2} + 88 a + 68\right)\cdot 173^{3} + \left(158 a^{2} + 166 a + 44\right)\cdot 173^{4} + \left(6 a^{2} + 111\right)\cdot 173^{5} + \left(76 a^{2} + 100 a + 54\right)\cdot 173^{6} + \left(100 a^{2} + 93 a + 117\right)\cdot 173^{7} + \left(56 a^{2} + 12 a + 150\right)\cdot 173^{8} + \left(31 a^{2} + 132 a + 112\right)\cdot 173^{9} + \left(114 a^{2} + 110 a + 23\right)\cdot 173^{10} + \left(79 a^{2} + 149 a + 127\right)\cdot 173^{11} + \left(95 a^{2} + 95 a + 94\right)\cdot 173^{12} + \left(33 a^{2} + 102 a + 130\right)\cdot 173^{13} + \left(150 a^{2} + 122 a + 55\right)\cdot 173^{14} + \left(129 a^{2} + 12 a + 106\right)\cdot 173^{15} + \left(86 a^{2} + 38 a + 97\right)\cdot 173^{16} + \left(165 a^{2} + 11 a + 18\right)\cdot 173^{17} + \left(40 a^{2} + 143 a + 67\right)\cdot 173^{18} + \left(39 a^{2} + 39 a + 99\right)\cdot 173^{19} +O\left(173^{ 20 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 96 a^{2} + 119 a + 122 + \left(76 a^{2} + 148 a + 21\right)\cdot 173 + \left(26 a^{2} + 49 a + 97\right)\cdot 173^{2} + \left(152 a^{2} + 25 a + 42\right)\cdot 173^{3} + \left(123 a^{2} + 57 a + 2\right)\cdot 173^{4} + \left(a^{2} + 170 a + 2\right)\cdot 173^{5} + \left(154 a^{2} + 137 a + 102\right)\cdot 173^{6} + \left(37 a^{2} + 162 a + 163\right)\cdot 173^{7} + \left(99 a^{2} + 41 a + 129\right)\cdot 173^{8} + \left(91 a^{2} + 23 a + 80\right)\cdot 173^{9} + \left(146 a^{2} + 133 a + 114\right)\cdot 173^{10} + \left(89 a^{2} + 51 a + 3\right)\cdot 173^{11} + \left(121 a^{2} + 6 a + 58\right)\cdot 173^{12} + \left(93 a^{2} + 91 a + 134\right)\cdot 173^{13} + \left(107 a^{2} + 135 a + 126\right)\cdot 173^{14} + \left(164 a^{2} + 118 a + 19\right)\cdot 173^{15} + \left(37 a^{2} + 74 a + 65\right)\cdot 173^{16} + \left(127 a^{2} + 100 a + 138\right)\cdot 173^{17} + \left(7 a^{2} + 122 a + 26\right)\cdot 173^{18} + \left(163 a^{2} + 7 a + 18\right)\cdot 173^{19} +O\left(173^{ 20 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 98 a^{2} + 64 a + 67 + \left(7 a^{2} + 166 a + 160\right)\cdot 173 + \left(46 a^{2} + 30 a + 7\right)\cdot 173^{2} + \left(134 a^{2} + 133 a + 134\right)\cdot 173^{3} + \left(105 a^{2} + 52 a + 35\right)\cdot 173^{4} + \left(15 a^{2} + 158 a + 78\right)\cdot 173^{5} + \left(34 a^{2} + 52 a + 57\right)\cdot 173^{6} + \left(108 a^{2} + 41 a + 84\right)\cdot 173^{7} + \left(83 a^{2} + 13 a + 51\right)\cdot 173^{8} + \left(128 a^{2} + 161 a + 72\right)\cdot 173^{9} + \left(32 a^{2} + 171 a + 20\right)\cdot 173^{10} + \left(86 a^{2} + 38 a + 114\right)\cdot 173^{11} + \left(58 a^{2} + 160 a + 31\right)\cdot 173^{12} + \left(92 a^{2} + 164 a + 17\right)\cdot 173^{13} + \left(159 a^{2} + 39 a + 23\right)\cdot 173^{14} + \left(34 a^{2} + 75 a + 135\right)\cdot 173^{15} + \left(31 a^{2} + 134 a + 113\right)\cdot 173^{16} + \left(157 a^{2} + 157 a + 120\right)\cdot 173^{17} + \left(91 a^{2} + 81 a + 23\right)\cdot 173^{18} + \left(85 a^{2} + 134 a + 30\right)\cdot 173^{19} +O\left(173^{ 20 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 152 a^{2} + 163 a + 139 + \left(88 a^{2} + 30 a + 95\right)\cdot 173 + \left(100 a^{2} + 92 a + 80\right)\cdot 173^{2} + \left(59 a^{2} + 14 a + 34\right)\cdot 173^{3} + \left(116 a^{2} + 63 a + 165\right)\cdot 173^{4} + \left(155 a^{2} + 17 a + 91\right)\cdot 173^{5} + \left(157 a^{2} + 155 a + 49\right)\cdot 173^{6} + \left(26 a^{2} + 141 a + 91\right)\cdot 173^{7} + \left(163 a^{2} + 117 a + 157\right)\cdot 173^{8} + \left(125 a^{2} + 161 a + 68\right)\cdot 173^{9} + \left(166 a^{2} + 40 a + 141\right)\cdot 173^{10} + \left(169 a^{2} + 82 a + 52\right)\cdot 173^{11} + \left(165 a^{2} + 6 a + 117\right)\cdot 173^{12} + \left(159 a^{2} + 90 a + 49\right)\cdot 173^{13} + \left(78 a^{2} + 170 a + 146\right)\cdot 173^{14} + \left(146 a^{2} + 151 a + 110\right)\cdot 173^{15} + \left(103 a^{2} + 136 a + 37\right)\cdot 173^{16} + \left(61 a^{2} + 87 a + 166\right)\cdot 173^{17} + \left(73 a^{2} + 141 a + 171\right)\cdot 173^{18} + \left(97 a^{2} + 30 a + 45\right)\cdot 173^{19} +O\left(173^{ 20 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2,3)(4,5,6)$ |
| $(4,5,6)(7,9,8)$ |
| $(1,3,2)$ |
| $(4,8,6,9,5,7)$ |
| $(1,8,3,9,2,7)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $9$ | $2$ | $(1,9)(2,8)(3,7)$ | $-1$ |
| $1$ | $3$ | $(1,3,2)(4,6,5)(7,8,9)$ | $3 \zeta_{3}$ |
| $1$ | $3$ | $(1,2,3)(4,5,6)(7,9,8)$ | $-3 \zeta_{3} - 3$ |
| $3$ | $3$ | $(1,2,3)(4,5,6)$ | $\zeta_{3} + 2$ |
| $3$ | $3$ | $(1,3,2)(4,6,5)$ | $-\zeta_{3} + 1$ |
| $3$ | $3$ | $(1,3,2)$ | $\zeta_{3} - 1$ |
| $3$ | $3$ | $(1,2,3)$ | $-\zeta_{3} - 2$ |
| $3$ | $3$ | $(1,3,2)(4,5,6)(7,8,9)$ | $-2 \zeta_{3} - 1$ |
| $3$ | $3$ | $(1,2,3)(4,6,5)(7,9,8)$ | $2 \zeta_{3} + 1$ |
| $6$ | $3$ | $(1,2,3)(7,8,9)$ | $0$ |
| $18$ | $3$ | $(1,6,7)(2,4,9)(3,5,8)$ | $0$ |
| $9$ | $6$ | $(1,8,3,9,2,7)(4,6,5)$ | $-1$ |
| $9$ | $6$ | $(1,7,2,9,3,8)(4,5,6)$ | $-1$ |
| $9$ | $6$ | $(4,8,6,9,5,7)$ | $-\zeta_{3}$ |
| $9$ | $6$ | $(4,7,5,9,6,8)$ | $\zeta_{3} + 1$ |
| $9$ | $6$ | $(1,2,3)(4,8)(5,7)(6,9)$ | $-\zeta_{3}$ |
| $9$ | $6$ | $(1,3,2)(4,8)(5,7)(6,9)$ | $\zeta_{3} + 1$ |
| $9$ | $6$ | $(1,2,3)(4,7,6,8,5,9)$ | $\zeta_{3} + 1$ |
| $9$ | $6$ | $(1,3,2)(4,9,5,8,6,7)$ | $-\zeta_{3}$ |
| $18$ | $9$ | $(1,4,8,3,6,9,2,5,7)$ | $0$ |
| $18$ | $9$ | $(1,8,6,2,7,4,3,9,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
