Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 127 }$ to precision 21.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$: $ x^{2} + 126 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 47 a + 83 + \left(4 a + 30\right)\cdot 127 + \left(12 a + 75\right)\cdot 127^{2} + \left(93 a + 42\right)\cdot 127^{3} + \left(110 a + 12\right)\cdot 127^{4} + \left(29 a + 78\right)\cdot 127^{5} + \left(91 a + 118\right)\cdot 127^{6} + \left(89 a + 36\right)\cdot 127^{7} + \left(3 a + 24\right)\cdot 127^{8} + \left(123 a + 4\right)\cdot 127^{9} + \left(44 a + 99\right)\cdot 127^{10} + \left(119 a + 26\right)\cdot 127^{11} + \left(75 a + 36\right)\cdot 127^{12} + \left(106 a + 102\right)\cdot 127^{13} + \left(117 a + 47\right)\cdot 127^{14} + \left(95 a + 71\right)\cdot 127^{15} + \left(118 a + 1\right)\cdot 127^{16} + \left(a + 107\right)\cdot 127^{17} + \left(31 a + 85\right)\cdot 127^{18} + \left(10 a + 111\right)\cdot 127^{19} + \left(94 a + 78\right)\cdot 127^{20} +O\left(127^{ 21 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 81 + 98\cdot 127 + 9\cdot 127^{2} + 55\cdot 127^{3} + 7\cdot 127^{4} + 65\cdot 127^{5} + 55\cdot 127^{6} + 63\cdot 127^{7} + 12\cdot 127^{8} + 87\cdot 127^{9} + 9\cdot 127^{10} + 82\cdot 127^{11} + 92\cdot 127^{12} + 123\cdot 127^{13} + 105\cdot 127^{14} + 10\cdot 127^{15} + 11\cdot 127^{16} + 92\cdot 127^{17} + 86\cdot 127^{18} + 109\cdot 127^{19} + 10\cdot 127^{20} +O\left(127^{ 21 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 80 a + 45 + \left(122 a + 96\right)\cdot 127 + \left(114 a + 51\right)\cdot 127^{2} + \left(33 a + 84\right)\cdot 127^{3} + \left(16 a + 114\right)\cdot 127^{4} + \left(97 a + 48\right)\cdot 127^{5} + \left(35 a + 8\right)\cdot 127^{6} + \left(37 a + 90\right)\cdot 127^{7} + \left(123 a + 102\right)\cdot 127^{8} + \left(3 a + 122\right)\cdot 127^{9} + \left(82 a + 27\right)\cdot 127^{10} + \left(7 a + 100\right)\cdot 127^{11} + \left(51 a + 90\right)\cdot 127^{12} + \left(20 a + 24\right)\cdot 127^{13} + \left(9 a + 79\right)\cdot 127^{14} + \left(31 a + 55\right)\cdot 127^{15} + \left(8 a + 125\right)\cdot 127^{16} + \left(125 a + 19\right)\cdot 127^{17} + \left(95 a + 41\right)\cdot 127^{18} + \left(116 a + 15\right)\cdot 127^{19} + \left(32 a + 48\right)\cdot 127^{20} +O\left(127^{ 21 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 47 + 28\cdot 127 + 117\cdot 127^{2} + 71\cdot 127^{3} + 119\cdot 127^{4} + 61\cdot 127^{5} + 71\cdot 127^{6} + 63\cdot 127^{7} + 114\cdot 127^{8} + 39\cdot 127^{9} + 117\cdot 127^{10} + 44\cdot 127^{11} + 34\cdot 127^{12} + 3\cdot 127^{13} + 21\cdot 127^{14} + 116\cdot 127^{15} + 115\cdot 127^{16} + 34\cdot 127^{17} + 40\cdot 127^{18} + 17\cdot 127^{19} + 116\cdot 127^{20} +O\left(127^{ 21 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 80 a + 3 + \left(122 a + 115\right)\cdot 127 + \left(114 a + 82\right)\cdot 127^{2} + \left(33 a + 123\right)\cdot 127^{3} + \left(16 a + 29\right)\cdot 127^{4} + \left(97 a + 124\right)\cdot 127^{5} + \left(35 a + 52\right)\cdot 127^{6} + \left(37 a + 35\right)\cdot 127^{7} + \left(123 a + 65\right)\cdot 127^{8} + \left(3 a + 123\right)\cdot 127^{9} + \left(82 a + 20\right)\cdot 127^{10} + \left(7 a + 101\right)\cdot 127^{11} + \left(51 a + 119\right)\cdot 127^{12} + \left(20 a + 5\right)\cdot 127^{13} + \left(9 a + 59\right)\cdot 127^{14} + \left(31 a + 49\right)\cdot 127^{15} + \left(8 a + 24\right)\cdot 127^{16} + \left(125 a + 117\right)\cdot 127^{17} + \left(95 a + 114\right)\cdot 127^{18} + \left(116 a + 90\right)\cdot 127^{19} + \left(32 a + 35\right)\cdot 127^{20} +O\left(127^{ 21 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 47 a + 125 + \left(4 a + 11\right)\cdot 127 + \left(12 a + 44\right)\cdot 127^{2} + \left(93 a + 3\right)\cdot 127^{3} + \left(110 a + 97\right)\cdot 127^{4} + \left(29 a + 2\right)\cdot 127^{5} + \left(91 a + 74\right)\cdot 127^{6} + \left(89 a + 91\right)\cdot 127^{7} + \left(3 a + 61\right)\cdot 127^{8} + \left(123 a + 3\right)\cdot 127^{9} + \left(44 a + 106\right)\cdot 127^{10} + \left(119 a + 25\right)\cdot 127^{11} + \left(75 a + 7\right)\cdot 127^{12} + \left(106 a + 121\right)\cdot 127^{13} + \left(117 a + 67\right)\cdot 127^{14} + \left(95 a + 77\right)\cdot 127^{15} + \left(118 a + 102\right)\cdot 127^{16} + \left(a + 9\right)\cdot 127^{17} + \left(31 a + 12\right)\cdot 127^{18} + \left(10 a + 36\right)\cdot 127^{19} + \left(94 a + 91\right)\cdot 127^{20} +O\left(127^{ 21 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 76 a + 26 + \left(115 a + 107\right)\cdot 127 + \left(38 a + 101\right)\cdot 127^{2} + \left(36 a + 64\right)\cdot 127^{3} + \left(95 a + 97\right)\cdot 127^{4} + \left(111 a + 118\right)\cdot 127^{5} + \left(88 a + 74\right)\cdot 127^{6} + \left(47 a + 20\right)\cdot 127^{7} + \left(7 a + 20\right)\cdot 127^{8} + \left(13 a + 124\right)\cdot 127^{9} + \left(22 a + 58\right)\cdot 127^{10} + \left(49 a + 113\right)\cdot 127^{11} + \left(50 a + 62\right)\cdot 127^{12} + \left(58 a + 59\right)\cdot 127^{13} + \left(52 a + 66\right)\cdot 127^{14} + \left(59 a + 123\right)\cdot 127^{15} + \left(116 a + 34\right)\cdot 127^{16} + \left(98 a + 72\right)\cdot 127^{17} + \left(124 a + 50\right)\cdot 127^{18} + \left(41 a + 41\right)\cdot 127^{19} + \left(117 a + 89\right)\cdot 127^{20} +O\left(127^{ 21 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 51 a + 102 + \left(11 a + 19\right)\cdot 127 + \left(88 a + 25\right)\cdot 127^{2} + \left(90 a + 62\right)\cdot 127^{3} + \left(31 a + 29\right)\cdot 127^{4} + \left(15 a + 8\right)\cdot 127^{5} + \left(38 a + 52\right)\cdot 127^{6} + \left(79 a + 106\right)\cdot 127^{7} + \left(119 a + 106\right)\cdot 127^{8} + \left(113 a + 2\right)\cdot 127^{9} + \left(104 a + 68\right)\cdot 127^{10} + \left(77 a + 13\right)\cdot 127^{11} + \left(76 a + 64\right)\cdot 127^{12} + \left(68 a + 67\right)\cdot 127^{13} + \left(74 a + 60\right)\cdot 127^{14} + \left(67 a + 3\right)\cdot 127^{15} + \left(10 a + 92\right)\cdot 127^{16} + \left(28 a + 54\right)\cdot 127^{17} + \left(2 a + 76\right)\cdot 127^{18} + \left(85 a + 85\right)\cdot 127^{19} + \left(9 a + 37\right)\cdot 127^{20} +O\left(127^{ 21 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,8,5)(4,7,6)$ |
| $(2,7,5)(4,8,6)$ |
| $(1,7,4)(2,3,8)$ |
| $(5,6)(7,8)$ |
| $(2,5)(4,6)(7,8)$ |
| $(1,8,4)(2,3,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,3)(2,4)(5,6)(7,8)$ |
$-4$ |
| $6$ |
$2$ |
$(1,3)(7,8)$ |
$0$ |
| $12$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$0$ |
| $24$ |
$2$ |
$(1,6)(2,4)(3,5)$ |
$0$ |
| $32$ |
$3$ |
$(1,7,4)(2,3,8)$ |
$1$ |
| $6$ |
$4$ |
$(1,2,3,4)(5,7,6,8)$ |
$0$ |
| $6$ |
$4$ |
$(1,5,3,6)(2,7,4,8)$ |
$0$ |
| $12$ |
$4$ |
$(2,6,4,5)$ |
$-2$ |
| $12$ |
$4$ |
$(1,5,3,6)(2,4)(7,8)$ |
$2$ |
| $32$ |
$6$ |
$(1,5,2,3,6,4)(7,8)$ |
$-1$ |
| $24$ |
$8$ |
$(1,7,4,5,3,8,2,6)$ |
$0$ |
| $24$ |
$8$ |
$(1,8,4,5,3,7,2,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
