Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 127 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$: $ x^{3} + 3 x + 124 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 71 + 44\cdot 127 + 20\cdot 127^{2} + 13\cdot 127^{3} + 78\cdot 127^{4} + 75\cdot 127^{5} + 93\cdot 127^{6} + 58\cdot 127^{7} + 95\cdot 127^{8} + 80\cdot 127^{9} +O\left(127^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 79 + 59\cdot 127 + 17\cdot 127^{2} + 38\cdot 127^{3} + 53\cdot 127^{4} + 4\cdot 127^{5} + 63\cdot 127^{6} + 113\cdot 127^{7} + 102\cdot 127^{8} + 90\cdot 127^{9} +O\left(127^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 104 + 22\cdot 127 + 89\cdot 127^{2} + 75\cdot 127^{3} + 122\cdot 127^{4} + 46\cdot 127^{5} + 97\cdot 127^{6} + 81\cdot 127^{7} + 55\cdot 127^{8} + 82\cdot 127^{9} +O\left(127^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 a^{2} + 6 a + 32 + \left(102 a^{2} + 2 a + 77\right)\cdot 127 + \left(116 a^{2} + 86 a + 106\right)\cdot 127^{2} + \left(69 a^{2} + 82 a + 12\right)\cdot 127^{3} + \left(43 a^{2} + 126 a + 87\right)\cdot 127^{4} + \left(4 a^{2} + 24 a + 8\right)\cdot 127^{5} + \left(112 a^{2} + 37 a + 97\right)\cdot 127^{6} + \left(24 a^{2} + 15 a + 49\right)\cdot 127^{7} + \left(107 a^{2} + 72 a + 87\right)\cdot 127^{8} + \left(54 a^{2} + 57 a + 109\right)\cdot 127^{9} +O\left(127^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 50 a^{2} + 114 a + 100 + \left(24 a^{2} + 18 a + 48\right)\cdot 127 + \left(73 a^{2} + 3 a + 19\right)\cdot 127^{2} + \left(69 a^{2} + 12 a + 12\right)\cdot 127^{3} + \left(22 a^{2} + 45 a + 45\right)\cdot 127^{4} + \left(110 a^{2} + 13 a + 93\right)\cdot 127^{5} + \left(111 a^{2} + 98 a + 96\right)\cdot 127^{6} + \left(31 a^{2} + 90 a + 63\right)\cdot 127^{7} + \left(24 a^{2} + 85 a + 48\right)\cdot 127^{8} + \left(82 a^{2} + 41 a + 37\right)\cdot 127^{9} +O\left(127^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 60 a^{2} + 94 a + 120 + \left(120 a^{2} + 66 a + 113\right)\cdot 127 + \left(107 a^{2} + 87 a + 88\right)\cdot 127^{2} + \left(56 a^{2} + 89 a + 113\right)\cdot 127^{3} + \left(24 a^{2} + 81 a + 48\right)\cdot 127^{4} + \left(22 a^{2} + 101 a + 44\right)\cdot 127^{5} + \left(35 a^{2} + 33 a + 70\right)\cdot 127^{6} + \left(74 a^{2} + 51 a + 21\right)\cdot 127^{7} + \left(38 a^{2} + 78 a + 77\right)\cdot 127^{8} + \left(75 a^{2} + 88 a + 23\right)\cdot 127^{9} +O\left(127^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 61 a^{2} + 7 a + 122 + 106 a\cdot 127 + \left(64 a^{2} + 37 a + 1\right)\cdot 127^{2} + \left(114 a^{2} + 32 a + 102\right)\cdot 127^{3} + \left(60 a^{2} + 82 a + 121\right)\cdot 127^{4} + \left(12 a^{2} + 88 a + 24\right)\cdot 127^{5} + \left(30 a^{2} + 118 a + 60\right)\cdot 127^{6} + \left(70 a^{2} + 20 a + 13\right)\cdot 127^{7} + \left(122 a^{2} + 96 a + 118\right)\cdot 127^{8} + \left(116 a^{2} + 27 a + 106\right)\cdot 127^{9} +O\left(127^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 70 a^{2} + 25 a + 13 + \left(76 a^{2} + 79 a + 26\right)\cdot 127 + \left(92 a^{2} + 15 a + 58\right)\cdot 127^{2} + \left(83 a^{2} + 108 a + 40\right)\cdot 127^{3} + \left(51 a^{2} + 87 a + 103\right)\cdot 127^{4} + \left(90 a^{2} + 4 a + 53\right)\cdot 127^{5} + \left(55 a^{2} + 65 a + 111\right)\cdot 127^{6} + \left(71 a^{2} + 51 a + 15\right)\cdot 127^{7} + \left(79 a^{2} + 59 a + 32\right)\cdot 127^{8} + \left(96 a^{2} + 85 a + 66\right)\cdot 127^{9} +O\left(127^{ 10 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 124 a^{2} + 8 a + 121 + \left(56 a^{2} + 108 a + 113\right)\cdot 127 + \left(53 a^{2} + 23 a + 106\right)\cdot 127^{2} + \left(113 a^{2} + 56 a + 99\right)\cdot 127^{3} + \left(50 a^{2} + 84 a + 101\right)\cdot 127^{4} + \left(14 a^{2} + 20 a + 28\right)\cdot 127^{5} + \left(36 a^{2} + 28 a + 72\right)\cdot 127^{6} + \left(108 a^{2} + 24 a + 89\right)\cdot 127^{7} + \left(8 a^{2} + 116 a + 17\right)\cdot 127^{8} + \left(82 a^{2} + 79 a + 37\right)\cdot 127^{9} +O\left(127^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,9)(2,8)(3,6)$ |
| $(1,7,9)(2,4,8)(3,5,6)$ |
| $(1,3,2)(4,7,5)(6,8,9)$ |
| $(2,3)(4,5)(6,8)$ |
| $(1,2,3)(6,8,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(2,3)(4,5)(6,8)$ | $0$ |
| $9$ | $2$ | $(1,9)(2,8)(3,6)$ | $2$ |
| $9$ | $2$ | $(1,9)(2,6)(3,8)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,3,2)(4,7,5)(6,8,9)$ | $-3$ |
| $6$ | $3$ | $(1,7,9)(2,4,8)(3,5,6)$ | $0$ |
| $6$ | $3$ | $(1,2,3)(4,7,5)$ | $0$ |
| $12$ | $3$ | $(1,6,5)(2,9,7)(3,8,4)$ | $0$ |
| $18$ | $6$ | $(1,7,9)(2,5,8,3,4,6)$ | $0$ |
| $18$ | $6$ | $(1,6,2,9,3,8)(4,7,5)$ | $-1$ |
| $18$ | $6$ | $(1,7,2,5,3,4)(6,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
