Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$: $ x^{3} + x + 188 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 71\cdot 193 + 112\cdot 193^{2} + 102\cdot 193^{3} + 170\cdot 193^{4} + 148\cdot 193^{5} + 167\cdot 193^{6} + 33\cdot 193^{7} + 170\cdot 193^{8} +O\left(193^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 38 + 52\cdot 193 + 15\cdot 193^{2} + 2\cdot 193^{3} + 109\cdot 193^{4} + 153\cdot 193^{5} + 12\cdot 193^{6} + 26\cdot 193^{7} + 136\cdot 193^{8} +O\left(193^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 161 + 63\cdot 193 + 184\cdot 193^{2} + 185\cdot 193^{3} + 163\cdot 193^{4} + 77\cdot 193^{5} + 188\cdot 193^{6} + 5\cdot 193^{7} + 83\cdot 193^{8} +O\left(193^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 a^{2} + 141 a + 108 + \left(109 a^{2} + 147 a + 175\right)\cdot 193 + \left(99 a^{2} + 83 a + 119\right)\cdot 193^{2} + \left(58 a^{2} + 52 a + 161\right)\cdot 193^{3} + \left(4 a^{2} + 170 a + 161\right)\cdot 193^{4} + \left(73 a^{2} + 167 a + 151\right)\cdot 193^{5} + \left(162 a^{2} + 156 a + 93\right)\cdot 193^{6} + \left(94 a^{2} + 137 a + 156\right)\cdot 193^{7} + \left(71 a^{2} + 4 a + 130\right)\cdot 193^{8} +O\left(193^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 60 a^{2} + 185 a + 5 + \left(178 a^{2} + 16 a + 18\right)\cdot 193 + \left(122 a^{2} + 19 a + 53\right)\cdot 193^{2} + \left(5 a^{2} + 146 a + 170\right)\cdot 193^{3} + \left(12 a^{2} + 159 a + 22\right)\cdot 193^{4} + \left(21 a^{2} + 157 a + 170\right)\cdot 193^{5} + \left(185 a^{2} + 58 a + 14\right)\cdot 193^{6} + \left(82 a^{2} + 120 a + 133\right)\cdot 193^{7} + \left(47 a^{2} + 88 a + 11\right)\cdot 193^{8} +O\left(193^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 72 a^{2} + 11 a + 142 + \left(141 a^{2} + 138 a + 132\right)\cdot 193 + \left(2 a^{2} + 122 a + 119\right)\cdot 193^{2} + \left(67 a^{2} + 114 a + 38\right)\cdot 193^{3} + \left(128 a^{2} + 153 a + 180\right)\cdot 193^{4} + \left(125 a^{2} + 121 a + 186\right)\cdot 193^{5} + \left(77 a^{2} + 124 a + 165\right)\cdot 193^{6} + \left(96 a^{2} + 31 a + 28\right)\cdot 193^{7} + \left(175 a^{2} + 28 a + 7\right)\cdot 193^{8} +O\left(193^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 100 a^{2} + 41 a + 32 + \left(135 a^{2} + 100 a\right)\cdot 193 + \left(90 a^{2} + 179 a + 114\right)\cdot 193^{2} + \left(67 a^{2} + 25 a + 167\right)\cdot 193^{3} + \left(60 a^{2} + 62 a + 134\right)\cdot 193^{4} + \left(187 a^{2} + 96 a + 163\right)\cdot 193^{5} + \left(145 a^{2} + 104 a + 82\right)\cdot 193^{6} + \left(a^{2} + 23 a + 94\right)\cdot 193^{7} + \left(139 a^{2} + 160 a + 111\right)\cdot 193^{8} +O\left(193^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 137 a^{2} + 11 a + 185 + \left(9 a^{2} + 96 a + 162\right)\cdot 193 + \left(162 a^{2} + 162 a + 14\right)\cdot 193^{2} + \left(20 a^{2} + 50 a + 116\right)\cdot 193^{3} + \left(56 a^{2} + 19 a + 116\right)\cdot 193^{4} + \left(184 a^{2} + 165 a + 21\right)\cdot 193^{5} + \left(6 a^{2} + 96 a + 89\right)\cdot 193^{6} + \left(181 a^{2} + 11 a + 5\right)\cdot 193^{7} + \left(60 a^{2} + 31 a + 85\right)\cdot 193^{8} +O\left(193^{ 9 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 189 a^{2} + 190 a + 91 + \left(4 a^{2} + 79 a + 95\right)\cdot 193 + \left(101 a^{2} + 11 a + 38\right)\cdot 193^{2} + \left(166 a^{2} + 189 a + 20\right)\cdot 193^{3} + \left(124 a^{2} + 13 a + 98\right)\cdot 193^{4} + \left(180 a^{2} + 63 a + 83\right)\cdot 193^{5} + \left(37 a + 149\right)\cdot 193^{6} + \left(122 a^{2} + 61 a + 94\right)\cdot 193^{7} + \left(84 a^{2} + 73 a + 36\right)\cdot 193^{8} +O\left(193^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3,2)(4,7,6)(5,8,9)$ |
| $(4,9)(5,7)(6,8)$ |
| $(4,6,7)(5,8,9)$ |
| $(2,3)(4,6)(8,9)$ |
| $(1,4,9)(2,6,8)(3,7,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(2,3)(4,7)(5,8)$ | $0$ |
| $9$ | $2$ | $(4,9)(5,7)(6,8)$ | $-2$ |
| $9$ | $2$ | $(1,7)(2,6)(3,4)(8,9)$ | $0$ |
| $2$ | $3$ | $(1,3,2)(4,7,6)(5,8,9)$ | $-3$ |
| $6$ | $3$ | $(1,9,6)(2,8,7)(3,5,4)$ | $0$ |
| $6$ | $3$ | $(1,3,2)(4,6,7)$ | $0$ |
| $12$ | $3$ | $(1,4,9)(2,6,8)(3,7,5)$ | $0$ |
| $18$ | $6$ | $(1,6,9)(2,4,8,3,7,5)$ | $0$ |
| $18$ | $6$ | $(1,3,2)(4,5,6,9,7,8)$ | $1$ |
| $18$ | $6$ | $(1,6,3,7,2,4)(8,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
