Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 191 }$ to precision 14.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 191 }$: $ x^{3} + 4 x + 172 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 96 + 59\cdot 191 + 24\cdot 191^{2} + 33\cdot 191^{3} + 171\cdot 191^{4} + 118\cdot 191^{5} + 89\cdot 191^{6} + 120\cdot 191^{7} + 56\cdot 191^{8} + 30\cdot 191^{9} + 23\cdot 191^{10} + 42\cdot 191^{11} + 63\cdot 191^{12} + 42\cdot 191^{13} +O\left(191^{ 14 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 109 + 31\cdot 191 + 170\cdot 191^{2} + 166\cdot 191^{3} + 180\cdot 191^{4} + 173\cdot 191^{5} + 61\cdot 191^{6} + 142\cdot 191^{7} + 177\cdot 191^{8} + 19\cdot 191^{9} + 34\cdot 191^{10} + 112\cdot 191^{11} + 178\cdot 191^{12} + 144\cdot 191^{13} +O\left(191^{ 14 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 177 + 99\cdot 191 + 187\cdot 191^{2} + 181\cdot 191^{3} + 29\cdot 191^{4} + 89\cdot 191^{5} + 39\cdot 191^{6} + 119\cdot 191^{7} + 147\cdot 191^{8} + 140\cdot 191^{9} + 133\cdot 191^{10} + 36\cdot 191^{11} + 140\cdot 191^{12} + 3\cdot 191^{13} +O\left(191^{ 14 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 a^{2} + 10 a + 186 + \left(74 a + 63\right)\cdot 191 + \left(185 a^{2} + 130 a + 111\right)\cdot 191^{2} + \left(124 a^{2} + 9 a + 78\right)\cdot 191^{3} + \left(92 a^{2} + 48 a + 183\right)\cdot 191^{4} + \left(3 a^{2} + 147 a + 72\right)\cdot 191^{5} + \left(110 a^{2} + 183 a + 102\right)\cdot 191^{6} + \left(26 a^{2} + 158 a + 134\right)\cdot 191^{7} + \left(178 a^{2} + 78 a + 156\right)\cdot 191^{8} + \left(106 a^{2} + 132 a + 157\right)\cdot 191^{9} + \left(167 a^{2} + 97 a + 64\right)\cdot 191^{10} + \left(24 a^{2} + 163 a + 66\right)\cdot 191^{11} + \left(93 a^{2} + 85 a + 57\right)\cdot 191^{12} + \left(170 a^{2} + 185 a + 136\right)\cdot 191^{13} +O\left(191^{ 14 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 a^{2} + 137 a + 141 + \left(160 a^{2} + 49 a + 108\right)\cdot 191 + \left(122 a^{2} + 143 a + 136\right)\cdot 191^{2} + \left(95 a^{2} + 123 a + 127\right)\cdot 191^{3} + \left(150 a^{2} + 61 a + 146\right)\cdot 191^{4} + \left(142 a^{2} + 68 a + 189\right)\cdot 191^{5} + \left(153 a^{2} + 180 a + 27\right)\cdot 191^{6} + \left(20 a^{2} + 13 a + 119\right)\cdot 191^{7} + \left(43 a^{2} + 44 a + 178\right)\cdot 191^{8} + \left(13 a^{2} + 119 a + 98\right)\cdot 191^{9} + \left(47 a^{2} + 70 a + 125\right)\cdot 191^{10} + \left(126 a^{2} + 159 a + 145\right)\cdot 191^{11} + \left(124 a^{2} + 84 a + 77\right)\cdot 191^{12} + \left(113 a^{2} + 72 a + 48\right)\cdot 191^{13} +O\left(191^{ 14 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 71 a^{2} + 188 a + 62 + \left(19 a^{2} + 17 a + 115\right)\cdot 191 + \left(54 a^{2} + 125 a + 80\right)\cdot 191^{2} + \left(44 a^{2} + 2 a + 54\right)\cdot 191^{3} + \left(140 a^{2} + 32 a + 119\right)\cdot 191^{4} + \left(152 a^{2} + 184 a + 152\right)\cdot 191^{5} + \left(30 a^{2} + 16 a + 145\right)\cdot 191^{6} + \left(42 a^{2} + 60 a + 48\right)\cdot 191^{7} + \left(44 a^{2} + 189 a + 54\right)\cdot 191^{8} + \left(23 a^{2} + 112 a + 189\right)\cdot 191^{9} + \left(184 a^{2} + 129 a + 108\right)\cdot 191^{10} + \left(10 a^{2} + 8 a + 156\right)\cdot 191^{11} + \left(65 a^{2} + 95 a + 109\right)\cdot 191^{12} + \left(107 a^{2} + 187 a + 31\right)\cdot 191^{13} +O\left(191^{ 14 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 138 a^{2} + 16 a + 177 + \left(68 a^{2} + 149 a + 55\right)\cdot 191 + \left(31 a^{2} + 146 a + 147\right)\cdot 191^{2} + \left(118 a^{2} + 96 a + 187\right)\cdot 191^{3} + \left(12 a^{2} + 17 a + 160\right)\cdot 191^{4} + \left(98 a^{2} + 34 a + 6\right)\cdot 191^{5} + \left(70 a^{2} + 154 a + 188\right)\cdot 191^{6} + \left(175 a^{2} + 87 a + 21\right)\cdot 191^{7} + \left(171 a^{2} + 92 a + 140\right)\cdot 191^{8} + \left(145 a^{2} + 46 a + 70\right)\cdot 191^{9} + \left(59 a^{2} + 8 a + 159\right)\cdot 191^{10} + \left(174 a^{2} + 32 a + 82\right)\cdot 191^{11} + \left(114 a^{2} + 186 a + 115\right)\cdot 191^{12} + \left(149 a^{2} + 20 a + 80\right)\cdot 191^{13} +O\left(191^{ 14 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 140 a^{2} + 44 a + 55 + \left(30 a^{2} + 67 a + 18\right)\cdot 191 + \left(74 a^{2} + 108 a + 134\right)\cdot 191^{2} + \left(161 a^{2} + 57 a + 175\right)\cdot 191^{3} + \left(138 a^{2} + 81 a + 51\right)\cdot 191^{4} + \left(44 a^{2} + 166 a + 119\right)\cdot 191^{5} + \left(118 a^{2} + 17 a + 60\right)\cdot 191^{6} + \left(143 a^{2} + 18 a + 128\right)\cdot 191^{7} + \left(160 a^{2} + 68 a + 46\right)\cdot 191^{8} + \left(70 a^{2} + 130 a + 125\right)\cdot 191^{9} + \left(167 a^{2} + 22 a\right)\cdot 191^{10} + \left(39 a^{2} + 59 a + 170\right)\cdot 191^{11} + \left(164 a^{2} + 20 a + 55\right)\cdot 191^{12} + \left(97 a^{2} + 124 a + 6\right)\cdot 191^{13} +O\left(191^{ 14 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 173 a^{2} + 178 a + 143 + \left(102 a^{2} + 23 a + 19\right)\cdot 191 + \left(105 a^{2} + 110 a + 154\right)\cdot 191^{2} + \left(28 a^{2} + 91 a + 139\right)\cdot 191^{3} + \left(38 a^{2} + 141 a + 101\right)\cdot 191^{4} + \left(131 a^{2} + 163 a + 31\right)\cdot 191^{5} + \left(89 a^{2} + 19 a + 48\right)\cdot 191^{6} + \left(164 a^{2} + 43 a + 120\right)\cdot 191^{7} + \left(165 a^{2} + 100 a + 187\right)\cdot 191^{8} + \left(21 a^{2} + 31 a + 121\right)\cdot 191^{9} + \left(138 a^{2} + 53 a + 113\right)\cdot 191^{10} + \left(5 a^{2} + 150 a + 142\right)\cdot 191^{11} + \left(11 a^{2} + 100 a + 156\right)\cdot 191^{12} + \left(125 a^{2} + 173 a + 78\right)\cdot 191^{13} +O\left(191^{ 14 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2,3)(6,9,7)$ |
| $(4,8,5)(6,9,7)$ |
| $(1,7,2,9,3,6)(4,5,8)$ |
| $(1,5,3,8,2,4)$ |
| $(6,9,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $9$ | $2$ | $(1,9)(2,6)(3,7)$ | $-1$ |
| $1$ | $3$ | $(1,2,3)(4,8,5)(6,7,9)$ | $3 \zeta_{3}$ |
| $1$ | $3$ | $(1,3,2)(4,5,8)(6,9,7)$ | $-3 \zeta_{3} - 3$ |
| $3$ | $3$ | $(1,3,2)(4,5,8)$ | $-2 \zeta_{3} - 1$ |
| $3$ | $3$ | $(1,2,3)(4,8,5)$ | $2 \zeta_{3} + 1$ |
| $3$ | $3$ | $(1,3,2)(4,5,8)(6,7,9)$ | $-\zeta_{3} - 2$ |
| $3$ | $3$ | $(1,2,3)(4,8,5)(6,9,7)$ | $\zeta_{3} - 1$ |
| $3$ | $3$ | $(6,7,9)$ | $\zeta_{3} + 2$ |
| $3$ | $3$ | $(6,9,7)$ | $-\zeta_{3} + 1$ |
| $6$ | $3$ | $(1,2,3)(6,9,7)$ | $0$ |
| $18$ | $3$ | $(1,4,9)(2,8,6)(3,5,7)$ | $0$ |
| $9$ | $6$ | $(1,7,2,9,3,6)(4,5,8)$ | $\zeta_{3} + 1$ |
| $9$ | $6$ | $(1,6,3,9,2,7)(4,8,5)$ | $-\zeta_{3}$ |
| $9$ | $6$ | $(1,5,3,8,2,4)$ | $-1$ |
| $9$ | $6$ | $(1,4,2,8,3,5)$ | $-1$ |
| $9$ | $6$ | $(1,8,3,4,2,5)(6,9,7)$ | $\zeta_{3} + 1$ |
| $9$ | $6$ | $(1,5,2,4,3,8)(6,7,9)$ | $-\zeta_{3}$ |
| $9$ | $6$ | $(1,5)(2,4)(3,8)(6,9,7)$ | $\zeta_{3} + 1$ |
| $9$ | $6$ | $(1,5)(2,4)(3,8)(6,7,9)$ | $-\zeta_{3}$ |
| $18$ | $9$ | $(1,7,4,3,6,5,2,9,8)$ | $0$ |
| $18$ | $9$ | $(1,4,6,2,8,7,3,5,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
