Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$: $ x^{3} + x + 188 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 33 + 42\cdot 193 + 54\cdot 193^{2} + 105\cdot 193^{3} + 29\cdot 193^{4} + 126\cdot 193^{5} + 89\cdot 193^{6} + 94\cdot 193^{7} +O\left(193^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 70 + 143\cdot 193 + 78\cdot 193^{2} + 23\cdot 193^{3} + 135\cdot 193^{4} + 188\cdot 193^{5} + 164\cdot 193^{6} + 155\cdot 193^{7} +O\left(193^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 90 + 7\cdot 193 + 60\cdot 193^{2} + 64\cdot 193^{3} + 28\cdot 193^{4} + 71\cdot 193^{5} + 131\cdot 193^{6} + 135\cdot 193^{7} +O\left(193^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 a^{2} + 47 a + 73 + \left(98 a^{2} + 47 a + 65\right)\cdot 193 + \left(85 a^{2} + 16 a + 121\right)\cdot 193^{2} + \left(9 a^{2} + 50 a + 70\right)\cdot 193^{3} + \left(143 a^{2} + 164 a + 95\right)\cdot 193^{4} + \left(91 a^{2} + 143 a + 125\right)\cdot 193^{5} + \left(137 a^{2} + 55 a + 91\right)\cdot 193^{6} + \left(4 a^{2} + 182 a + 67\right)\cdot 193^{7} +O\left(193^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 a^{2} + 8 a + 150 + \left(22 a^{2} + 6 a + 14\right)\cdot 193 + \left(143 a^{2} + 150 a + 31\right)\cdot 193^{2} + \left(86 a^{2} + 47 a + 122\right)\cdot 193^{3} + \left(108 a^{2} + 173 a + 136\right)\cdot 193^{4} + \left(4 a^{2} + 21 a + 131\right)\cdot 193^{5} + \left(155 a^{2} + 170 a + 167\right)\cdot 193^{6} + \left(161 a^{2} + 22 a + 107\right)\cdot 193^{7} +O\left(193^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 53 a^{2} + 58 a + 164 + \left(144 a^{2} + 53 a + 31\right)\cdot 193 + \left(182 a^{2} + 6 a + 186\right)\cdot 193^{2} + \left(a^{2} + 49 a + 129\right)\cdot 193^{3} + \left(6 a^{2} + 139 a + 132\right)\cdot 193^{4} + \left(10 a^{2} + 86 a + 6\right)\cdot 193^{5} + \left(113 a^{2} + 143 a + 11\right)\cdot 193^{6} + \left(93 a^{2} + 133 a + 191\right)\cdot 193^{7} +O\left(193^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 127 a^{2} + 88 a + 149 + \left(143 a^{2} + 92 a + 95\right)\cdot 193 + \left(117 a^{2} + 170 a + 78\right)\cdot 193^{2} + \left(181 a^{2} + 93 a + 185\right)\cdot 193^{3} + \left(43 a^{2} + 82 a + 157\right)\cdot 193^{4} + \left(91 a^{2} + 155 a + 60\right)\cdot 193^{5} + \left(135 a^{2} + 186 a + 90\right)\cdot 193^{6} + \left(94 a^{2} + 69 a + 127\right)\cdot 193^{7} +O\left(193^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 175 a^{2} + 92 a + 181 + \left(60 a^{2} + 117 a + 104\right)\cdot 193 + \left(89 a^{2} + 60 a + 59\right)\cdot 193^{2} + \left(49 a^{2} + 59 a + 97\right)\cdot 193^{3} + \left(187 a^{2} + 156 a + 60\right)\cdot 193^{4} + \left(168 a^{2} + 73 a + 48\right)\cdot 193^{5} + \left(108 a^{2} + 128 a + 8\right)\cdot 193^{6} + \left(65 a^{2} + 127 a + 108\right)\cdot 193^{7} +O\left(193^{ 8 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 179 a^{2} + 93 a + 55 + \left(109 a^{2} + 69 a + 73\right)\cdot 193 + \left(153 a^{2} + 175 a + 102\right)\cdot 193^{2} + \left(56 a^{2} + 85 a + 166\right)\cdot 193^{3} + \left(90 a^{2} + 56 a + 188\right)\cdot 193^{4} + \left(19 a^{2} + 97 a + 12\right)\cdot 193^{5} + \left(122 a^{2} + 87 a + 17\right)\cdot 193^{6} + \left(158 a^{2} + 42 a + 170\right)\cdot 193^{7} +O\left(193^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(4,9)(5,6)(7,8)$ |
| $(1,3,2)(4,6,7)(5,8,9)$ |
| $(1,5,6)(2,9,4)(3,8,7)$ |
| $(2,3)(4,6)(5,9)$ |
| $(4,6,7)(5,9,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(2,3)(4,7)(5,8)$ | $0$ |
| $9$ | $2$ | $(4,9)(5,6)(7,8)$ | $2$ |
| $9$ | $2$ | $(1,8)(2,9)(3,5)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,3,2)(4,6,7)(5,8,9)$ | $-3$ |
| $6$ | $3$ | $(1,6,9)(2,4,8)(3,7,5)$ | $0$ |
| $6$ | $3$ | $(1,3,2)(5,9,8)$ | $0$ |
| $12$ | $3$ | $(1,5,6)(2,9,4)(3,8,7)$ | $0$ |
| $18$ | $6$ | $(1,9,6)(2,5,4,3,8,7)$ | $0$ |
| $18$ | $6$ | $(1,3,2)(4,5,7,9,6,8)$ | $-1$ |
| $18$ | $6$ | $(1,9,3,8,2,5)(4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
