Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 139 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 139 }$: $ x^{3} + 6 x + 137 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 80\cdot 139 + 82\cdot 139^{2} + 118\cdot 139^{3} + 127\cdot 139^{4} + 81\cdot 139^{5} + 29\cdot 139^{6} + 47\cdot 139^{7} +O\left(139^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 129 + 43\cdot 139 + 95\cdot 139^{2} + 127\cdot 139^{3} + 107\cdot 139^{4} + 61\cdot 139^{5} + 133\cdot 139^{6} + 79\cdot 139^{7} +O\left(139^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 136 + 14\cdot 139 + 100\cdot 139^{2} + 31\cdot 139^{3} + 42\cdot 139^{4} + 134\cdot 139^{5} + 114\cdot 139^{6} + 11\cdot 139^{7} +O\left(139^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 a^{2} + 131 a + 32 + \left(60 a^{2} + 123 a + 101\right)\cdot 139 + \left(a^{2} + 39 a + 5\right)\cdot 139^{2} + \left(47 a^{2} + 34 a + 49\right)\cdot 139^{3} + \left(20 a^{2} + 64 a + 81\right)\cdot 139^{4} + \left(84 a^{2} + 56 a + 58\right)\cdot 139^{5} + \left(90 a^{2} + 26 a + 84\right)\cdot 139^{6} + \left(34 a^{2} + 115 a + 138\right)\cdot 139^{7} +O\left(139^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 a^{2} + 50 a + 100 + \left(14 a^{2} + 66 a + 56\right)\cdot 139 + \left(27 a^{2} + 16 a + 108\right)\cdot 139^{2} + \left(41 a^{2} + 118 a + 25\right)\cdot 139^{3} + \left(42 a^{2} + 14 a + 30\right)\cdot 139^{4} + \left(65 a^{2} + 41 a + 122\right)\cdot 139^{5} + \left(61 a^{2} + 115 a + 106\right)\cdot 139^{6} + \left(134 a^{2} + 88 a + 120\right)\cdot 139^{7} +O\left(139^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 a^{2} + 74 a + 9 + \left(97 a^{2} + 89 a + 111\right)\cdot 139 + \left(22 a^{2} + 134 a + 90\right)\cdot 139^{2} + \left(117 a^{2} + 31 a + 51\right)\cdot 139^{3} + \left(5 a^{2} + 136 a + 23\right)\cdot 139^{4} + \left(118 a^{2} + 118 a + 55\right)\cdot 139^{5} + \left(103 a^{2} + 56 a + 137\right)\cdot 139^{6} + \left(137 a^{2} + 47 a + 133\right)\cdot 139^{7} +O\left(139^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 58 a^{2} + 81 a + 93 + \left(94 a^{2} + 127 a + 99\right)\cdot 139 + \left(115 a^{2} + 124 a + 45\right)\cdot 139^{2} + \left(41 a^{2} + 38 a + 28\right)\cdot 139^{3} + \left(10 a^{2} + 130 a + 41\right)\cdot 139^{4} + \left(18 a^{2} + 120 a + 72\right)\cdot 139^{5} + \left(67 a^{2} + 29 a + 129\right)\cdot 139^{6} + \left(65 a^{2} + 137 a + 122\right)\cdot 139^{7} +O\left(139^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 73 a^{2} + 66 a + 14 + \left(123 a^{2} + 26 a + 77\right)\cdot 139 + \left(21 a^{2} + 113 a + 87\right)\cdot 139^{2} + \left(50 a^{2} + 65 a + 61\right)\cdot 139^{3} + \left(108 a^{2} + 83 a + 16\right)\cdot 139^{4} + \left(36 a^{2} + 100 a + 8\right)\cdot 139^{5} + \left(120 a^{2} + 82 a + 64\right)\cdot 139^{6} + \left(38 a^{2} + 25 a + 16\right)\cdot 139^{7} +O\left(139^{ 8 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 77 a^{2} + 15 a + 30 + \left(27 a^{2} + 122 a + 110\right)\cdot 139 + \left(89 a^{2} + 126 a + 78\right)\cdot 139^{2} + \left(119 a^{2} + 127 a + 61\right)\cdot 139^{3} + \left(90 a^{2} + 126 a + 85\right)\cdot 139^{4} + \left(94 a^{2} + 117 a + 100\right)\cdot 139^{5} + \left(112 a^{2} + 105 a + 33\right)\cdot 139^{6} + \left(5 a^{2} + 2 a + 23\right)\cdot 139^{7} +O\left(139^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,9,8)(2,6,4)(3,5,7)$ |
| $(1,3)(4,7)(5,6)$ |
| $(1,3,2)(4,8,7)(5,6,9)$ |
| $(4,6)(5,7)(8,9)$ |
| $(4,8,7)(5,9,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,3)(6,9)(7,8)$ | $0$ |
| $9$ | $2$ | $(4,6)(5,7)(8,9)$ | $-2$ |
| $9$ | $2$ | $(1,6)(2,9)(3,5)(4,7)$ | $0$ |
| $2$ | $3$ | $(1,3,2)(4,8,7)(5,6,9)$ | $-3$ |
| $6$ | $3$ | $(1,8,6)(2,4,5)(3,7,9)$ | $0$ |
| $6$ | $3$ | $(1,3,2)(5,9,6)$ | $0$ |
| $12$ | $3$ | $(1,9,8)(2,6,4)(3,5,7)$ | $0$ |
| $18$ | $6$ | $(1,9,8,3,6,7)(2,5,4)$ | $0$ |
| $18$ | $6$ | $(1,3,2)(4,9,7,6,8,5)$ | $1$ |
| $18$ | $6$ | $(1,9,3,6,2,5)(4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
