Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 127 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$: $ x^{3} + 3 x + 124 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 26\cdot 127 + 65\cdot 127^{2} + 10\cdot 127^{3} + 41\cdot 127^{4} + 23\cdot 127^{5} + 71\cdot 127^{6} + 106\cdot 127^{7} + 110\cdot 127^{8} + 10\cdot 127^{9} + 37\cdot 127^{10} + 34\cdot 127^{11} +O\left(127^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 58 + 90\cdot 127 + 72\cdot 127^{2} + 62\cdot 127^{3} + 102\cdot 127^{4} + 5\cdot 127^{5} + 99\cdot 127^{6} + 80\cdot 127^{7} + 49\cdot 127^{8} + 73\cdot 127^{9} + 118\cdot 127^{10} + 25\cdot 127^{11} +O\left(127^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 115 + 93\cdot 127 + 5\cdot 127^{2} + 76\cdot 127^{3} + 95\cdot 127^{4} + 92\cdot 127^{5} + 81\cdot 127^{6} + 10\cdot 127^{7} + 75\cdot 127^{8} + 70\cdot 127^{9} + 5\cdot 127^{10} + 33\cdot 127^{11} +O\left(127^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 a^{2} + 11 a + 74 + \left(17 a^{2} + 89 a + 121\right)\cdot 127 + \left(112 a^{2} + 90 a + 40\right)\cdot 127^{2} + \left(92 a^{2} + 101 a + 107\right)\cdot 127^{3} + \left(52 a^{2} + 82 a + 110\right)\cdot 127^{4} + \left(26 a^{2} + 64 a + 2\right)\cdot 127^{5} + \left(41 a^{2} + 35 a + 13\right)\cdot 127^{6} + \left(117 a^{2} + 80 a + 57\right)\cdot 127^{7} + \left(95 a^{2} + 108 a + 61\right)\cdot 127^{8} + \left(107 a^{2} + 28 a + 29\right)\cdot 127^{9} + \left(125 a^{2} + 44 a + 36\right)\cdot 127^{10} + \left(27 a^{2} + 83 a + 93\right)\cdot 127^{11} +O\left(127^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 36 a^{2} + 93 a + 100 + \left(28 a^{2} + 14 a + 16\right)\cdot 127 + \left(19 a^{2} + 78 a + 109\right)\cdot 127^{2} + \left(9 a^{2} + 67 a + 66\right)\cdot 127^{3} + \left(117 a^{2} + 70 a + 112\right)\cdot 127^{4} + \left(65 a^{2} + 13 a + 81\right)\cdot 127^{5} + \left(22 a^{2} + 19 a + 102\right)\cdot 127^{6} + \left(99 a^{2} + 84 a + 20\right)\cdot 127^{7} + \left(122 a^{2} + 2 a + 115\right)\cdot 127^{8} + \left(65 a^{2} + 72 a + 72\right)\cdot 127^{9} + \left(101 a^{2} + 103 a + 114\right)\cdot 127^{10} + \left(100 a^{2} + 111\right)\cdot 127^{11} +O\left(127^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 a^{2} + 70 a + 56 + \left(125 a^{2} + 36 a + 93\right)\cdot 127 + \left(43 a^{2} + 117 a + 11\right)\cdot 127^{2} + \left(110 a^{2} + 112 a + 80\right)\cdot 127^{3} + \left(49 a^{2} + 105 a + 14\right)\cdot 127^{4} + \left(97 a^{2} + 111 a + 77\right)\cdot 127^{5} + \left(95 a^{2} + 80 a + 7\right)\cdot 127^{6} + \left(12 a^{2} + 56 a + 10\right)\cdot 127^{7} + \left(21 a^{2} + 68 a + 94\right)\cdot 127^{8} + \left(93 a^{2} + 98 a + 66\right)\cdot 127^{9} + \left(50 a^{2} + 109 a + 51\right)\cdot 127^{10} + \left(64 a^{2} + a + 60\right)\cdot 127^{11} +O\left(127^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 68 a^{2} + 23 a + 37 + \left(81 a^{2} + 23 a + 123\right)\cdot 127 + \left(122 a^{2} + 85 a + 61\right)\cdot 127^{2} + \left(24 a^{2} + 84 a + 98\right)\cdot 127^{3} + \left(84 a^{2} + 100 a + 46\right)\cdot 127^{4} + \left(34 a^{2} + 48 a + 19\right)\cdot 127^{5} + \left(63 a^{2} + 72 a + 57\right)\cdot 127^{6} + \left(37 a^{2} + 89 a + 24\right)\cdot 127^{7} + \left(35 a^{2} + 15 a + 67\right)\cdot 127^{8} + \left(80 a^{2} + 26 a + 101\right)\cdot 127^{9} + \left(26 a^{2} + 106 a + 91\right)\cdot 127^{10} + \left(125 a^{2} + 42 a + 33\right)\cdot 127^{11} +O\left(127^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 100 a^{2} + 27 a + 29 + \left(53 a^{2} + 3 a + 77\right)\cdot 127 + \left(3 a^{2} + 114 a + 57\right)\cdot 127^{2} + \left(55 a^{2} + 40 a + 96\right)\cdot 127^{3} + \left(35 a^{2} + 50 a + 112\right)\cdot 127^{4} + \left(74 a^{2} + 47 a + 30\right)\cdot 127^{5} + \left(72 a^{2} + 103 a + 88\right)\cdot 127^{6} + \left(4 a^{2} + 126 a + 120\right)\cdot 127^{7} + \left(10 a^{2} + 9 a + 71\right)\cdot 127^{8} + \left(a^{2} + 53 a + 9\right)\cdot 127^{9} + \left(22 a^{2} + 54 a + 121\right)\cdot 127^{10} + \left(61 a^{2} + 40 a + 53\right)\cdot 127^{11} +O\left(127^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 104 a^{2} + 30 a + 37 + \left(74 a^{2} + 87 a + 119\right)\cdot 127 + \left(79 a^{2} + 22 a + 82\right)\cdot 127^{2} + \left(88 a^{2} + 100 a + 36\right)\cdot 127^{3} + \left(41 a^{2} + 97 a + 125\right)\cdot 127^{4} + \left(82 a^{2} + 94 a + 46\right)\cdot 127^{5} + \left(85 a^{2} + 69 a + 114\right)\cdot 127^{6} + \left(109 a^{2} + 70 a + 76\right)\cdot 127^{7} + \left(95 a^{2} + 48 a + 116\right)\cdot 127^{8} + \left(32 a^{2} + 102 a + 72\right)\cdot 127^{9} + \left(54 a^{2} + 89 a + 58\right)\cdot 127^{10} + \left(a^{2} + 84 a + 61\right)\cdot 127^{11} +O\left(127^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3,2)(6,8,9)$ |
| $(1,3,2)(4,5,7)$ |
| $(1,6,4)(2,8,5)(3,9,7)$ |
| $(1,8)(2,6)(3,9)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,8)(2,6)(3,9)(4,5)$ |
$0$ |
| $2$ |
$3$ |
$(1,2,3)(4,5,7)(6,8,9)$ |
$-3$ |
| $3$ |
$3$ |
$(1,3,2)(4,5,7)$ |
$0$ |
| $3$ |
$3$ |
$(1,2,3)(4,7,5)$ |
$0$ |
| $6$ |
$3$ |
$(1,6,4)(2,8,5)(3,9,7)$ |
$0$ |
| $6$ |
$3$ |
$(1,6,5)(2,8,7)(3,9,4)$ |
$0$ |
| $6$ |
$3$ |
$(1,5,6)(2,7,8)(3,4,9)$ |
$0$ |
| $9$ |
$6$ |
$(1,8,3,9,2,6)(4,7)$ |
$0$ |
| $9$ |
$6$ |
$(1,6,2,9,3,8)(4,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
