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:
| $r_{ 1 }$ |
$=$ |
$ 17 + 4\cdot 173 + 144\cdot 173^{2} + 154\cdot 173^{3} + 23\cdot 173^{4} + 71\cdot 173^{5} + 39\cdot 173^{6} + 73\cdot 173^{7} + 165\cdot 173^{8} + 155\cdot 173^{9} +O\left(173^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 77\cdot 173 + 10\cdot 173^{2} + 95\cdot 173^{3} + 40\cdot 173^{4} + 148\cdot 173^{5} + 171\cdot 173^{6} + 17\cdot 173^{7} + 116\cdot 173^{8} + 46\cdot 173^{9} +O\left(173^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 123 + 55\cdot 173 + 48\cdot 173^{2} + 20\cdot 173^{3} + 124\cdot 173^{4} + 55\cdot 173^{5} + 168\cdot 173^{6} + 6\cdot 173^{7} + 70\cdot 173^{8} + 44\cdot 173^{9} +O\left(173^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 a^{2} + 80 a + 50 + \left(129 a^{2} + 144 a + 98\right)\cdot 173 + \left(34 a^{2} + 54 a + 3\right)\cdot 173^{2} + \left(157 a^{2} + 25 a + 88\right)\cdot 173^{3} + \left(70 a^{2} + 156 a + 40\right)\cdot 173^{4} + \left(13 a^{2} + 156 a + 40\right)\cdot 173^{5} + \left(12 a^{2} + 92 a + 4\right)\cdot 173^{6} + \left(90 a^{2} + 41 a + 42\right)\cdot 173^{7} + \left(155 a^{2} + 97 a + 139\right)\cdot 173^{8} + \left(33 a^{2} + 128 a + 117\right)\cdot 173^{9} +O\left(173^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a^{2} + 40 a + 125 + \left(15 a^{2} + 33 a + 61\right)\cdot 173 + \left(135 a^{2} + 66 a + 137\right)\cdot 173^{2} + \left(93 a^{2} + 127 a + 118\right)\cdot 173^{3} + \left(29 a^{2} + 68 a + 100\right)\cdot 173^{4} + \left(143 a^{2} + 126 a + 155\right)\cdot 173^{5} + \left(24 a^{2} + 108 a + 78\right)\cdot 173^{6} + \left(162 a^{2} + 120 a + 80\right)\cdot 173^{7} + \left(40 a^{2} + 12 a + 159\right)\cdot 173^{8} + \left(26 a^{2} + 93 a + 49\right)\cdot 173^{9} +O\left(173^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 64 a^{2} + 13 a + 50 + \left(117 a^{2} + 69 a + 69\right)\cdot 173 + \left(84 a^{2} + 117 a + 30\right)\cdot 173^{2} + \left(2 a^{2} + 54 a + 150\right)\cdot 173^{3} + \left(51 a^{2} + 168 a + 116\right)\cdot 173^{4} + \left(157 a^{2} + 52 a + 95\right)\cdot 173^{5} + \left(49 a^{2} + 30 a + 9\right)\cdot 173^{6} + \left(135 a^{2} + 21 a + 168\right)\cdot 173^{7} + \left(8 a^{2} + 101 a + 77\right)\cdot 173^{8} + \left(125 a^{2} + 98 a + 69\right)\cdot 173^{9} +O\left(173^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 113 a^{2} + 44 a + \left(72 a^{2} + 125\right)\cdot 173 + \left(114 a^{2} + 61 a + 127\right)\cdot 173^{2} + \left(126 a^{2} + 12 a + 142\right)\cdot 173^{3} + \left(83 a^{2} + 113 a + 102\right)\cdot 173^{4} + \left(144 a^{2} + 35 a + 78\right)\cdot 173^{5} + \left(99 a^{2} + 171 a + 18\right)\cdot 173^{6} + \left(19 a^{2} + 81 a + 129\right)\cdot 173^{7} + \left(172 a^{2} + 3 a + 122\right)\cdot 173^{8} + \left(102 a^{2} + 155 a + 97\right)\cdot 173^{9} +O\left(173^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 144 a^{2} + 53 a + 116 + \left(28 a^{2} + 168 a + 137\right)\cdot 173 + \left(3 a^{2} + 51 a + 76\right)\cdot 173^{2} + \left(95 a^{2} + 20 a + 120\right)\cdot 173^{3} + \left(72 a^{2} + 121 a + 42\right)\cdot 173^{4} + \left(16 a^{2} + 62 a + 44\right)\cdot 173^{5} + \left(136 a^{2} + 144 a + 54\right)\cdot 173^{6} + \left(93 a^{2} + 10 a + 162\right)\cdot 173^{7} + \left(149 a^{2} + 63 a + 15\right)\cdot 173^{8} + \left(112 a^{2} + 124 a + 50\right)\cdot 173^{9} +O\left(173^{ 10 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 169 a^{2} + 116 a + 17 + \left(155 a^{2} + 103 a + 63\right)\cdot 173 + \left(146 a^{2} + 167 a + 113\right)\cdot 173^{2} + \left(43 a^{2} + 105 a + 147\right)\cdot 173^{3} + \left(38 a^{2} + 64 a + 99\right)\cdot 173^{4} + \left(44 a^{2} + 84 a + 2\right)\cdot 173^{5} + \left(23 a^{2} + 144 a + 147\right)\cdot 173^{6} + \left(18 a^{2} + 69 a + 11\right)\cdot 173^{7} + \left(165 a^{2} + 68 a + 171\right)\cdot 173^{8} + \left(117 a^{2} + 92 a + 59\right)\cdot 173^{9} +O\left(173^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(4,5,8)(6,9,7)$ |
| $(1,5,7)(2,8,9)(3,4,6)$ |
| $(1,2)(4,5)(6,7)$ |
| $(1,2,3)(4,5,8)(6,7,9)$ |
| $(4,6)(5,7)(8,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,2)(4,8)(7,9)$ |
$0$ |
| $9$ |
$2$ |
$(4,6)(5,7)(8,9)$ |
$-2$ |
| $9$ |
$2$ |
$(1,5)(2,4)(3,8)(6,7)$ |
$0$ |
| $2$ |
$3$ |
$(1,2,3)(4,5,8)(6,7,9)$ |
$-3$ |
| $6$ |
$3$ |
$(1,7,8)(2,9,4)(3,6,5)$ |
$0$ |
| $6$ |
$3$ |
$(1,3,2)(4,5,8)$ |
$0$ |
| $12$ |
$3$ |
$(1,5,7)(2,8,9)(3,4,6)$ |
$0$ |
| $18$ |
$6$ |
$(1,4,7,2,8,9)(3,5,6)$ |
$0$ |
| $18$ |
$6$ |
$(1,2,3)(4,7,8,6,5,9)$ |
$1$ |
| $18$ |
$6$ |
$(1,4,3,5,2,8)(6,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
