Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 13.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $ x^{3} + 3 x + 99 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 a^{2} + 70 a + 1 + \left(97 a^{2} + 77 a + 13\right)\cdot 101 + \left(8 a^{2} + 78 a + 7\right)\cdot 101^{2} + \left(52 a^{2} + 66 a + 2\right)\cdot 101^{3} + \left(3 a^{2} + 67 a + 21\right)\cdot 101^{4} + \left(4 a^{2} + 73 a + 95\right)\cdot 101^{5} + \left(51 a^{2} + 60 a + 40\right)\cdot 101^{6} + \left(97 a^{2} + 68 a + 24\right)\cdot 101^{7} + \left(29 a^{2} + 83 a + 64\right)\cdot 101^{8} + \left(18 a^{2} + 34 a + 71\right)\cdot 101^{9} + \left(87 a^{2} + 94 a + 9\right)\cdot 101^{10} + \left(34 a^{2} + 80 a + 89\right)\cdot 101^{11} + \left(99 a^{2} + 49 a + 56\right)\cdot 101^{12} +O\left(101^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 98 + 87\cdot 101 + 44\cdot 101^{2} + 91\cdot 101^{3} + 46\cdot 101^{4} + 56\cdot 101^{5} + 29\cdot 101^{7} + 63\cdot 101^{8} + 15\cdot 101^{9} + 26\cdot 101^{10} + 7\cdot 101^{12} +O\left(101^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 a^{2} + 83 a + 17 + \left(76 a^{2} + 61 a + 72\right)\cdot 101 + \left(56 a^{2} + 77 a + 1\right)\cdot 101^{2} + \left(32 a^{2} + 18 a + 64\right)\cdot 101^{3} + \left(16 a^{2} + 12 a + 46\right)\cdot 101^{4} + \left(82 a^{2} + 93 a + 49\right)\cdot 101^{5} + \left(44 a^{2} + 82 a + 28\right)\cdot 101^{6} + \left(58 a^{2} + 42 a + 47\right)\cdot 101^{7} + \left(26 a^{2} + 19 a + 57\right)\cdot 101^{8} + \left(44 a^{2} + 87 a + 22\right)\cdot 101^{9} + \left(13 a^{2} + 62 a + 64\right)\cdot 101^{10} + \left(27 a^{2} + 39 a + 73\right)\cdot 101^{11} + \left(88 a^{2} + 77 a + 34\right)\cdot 101^{12} +O\left(101^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 55 + 72\cdot 101 + 51\cdot 101^{2} + 32\cdot 101^{3} + 90\cdot 101^{4} + 63\cdot 101^{5} + 89\cdot 101^{6} + 26\cdot 101^{7} + 82\cdot 101^{8} + 78\cdot 101^{9} + 51\cdot 101^{10} + 52\cdot 101^{11} + 84\cdot 101^{12} +O\left(101^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 + 29\cdot 101 + 79\cdot 101^{2} + 28\cdot 101^{3} + 3\cdot 101^{4} + 86\cdot 101^{5} + 83\cdot 101^{6} + 55\cdot 101^{7} + 85\cdot 101^{8} + 23\cdot 101^{9} + 57\cdot 101^{10} + 12\cdot 101^{11} + 91\cdot 101^{12} +O\left(101^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 a^{2} + 44 a + 67 + \left(37 a^{2} + 72 a + 57\right)\cdot 101 + \left(88 a^{2} + 25 a + 61\right)\cdot 101^{2} + \left(83 a^{2} + 46 a + 50\right)\cdot 101^{3} + \left(67 a^{2} + 78 a + 7\right)\cdot 101^{4} + \left(16 a^{2} + 93 a + 12\right)\cdot 101^{5} + \left(26 a^{2} + 18 a + 89\right)\cdot 101^{6} + \left(16 a^{2} + 2 a + 64\right)\cdot 101^{7} + \left(27 a^{2} + 27 a + 6\right)\cdot 101^{8} + \left(60 a^{2} + 94 a + 46\right)\cdot 101^{9} + \left(89 a^{2} + 80 a + 29\right)\cdot 101^{10} + \left(19 a^{2} + 17 a + 32\right)\cdot 101^{11} + \left(28 a^{2} + 58 a + 36\right)\cdot 101^{12} +O\left(101^{ 13 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 79 a^{2} + 49 a + 44 + \left(28 a^{2} + 62 a + 78\right)\cdot 101 + \left(35 a^{2} + 45 a + 59\right)\cdot 101^{2} + \left(16 a^{2} + 15 a + 31\right)\cdot 101^{3} + \left(81 a^{2} + 21 a + 75\right)\cdot 101^{4} + \left(14 a^{2} + 35 a + 15\right)\cdot 101^{5} + \left(5 a^{2} + 58 a + 50\right)\cdot 101^{6} + \left(46 a^{2} + 90 a + 22\right)\cdot 101^{7} + \left(44 a^{2} + 98 a + 93\right)\cdot 101^{8} + \left(38 a^{2} + 79 a + 10\right)\cdot 101^{9} + \left(44 a + 38\right)\cdot 101^{10} + \left(39 a^{2} + 81 a + 97\right)\cdot 101^{11} + \left(14 a^{2} + 74 a + 88\right)\cdot 101^{12} +O\left(101^{ 13 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 74 a^{2} + 74 + \left(73 a^{2} + 44 a + 29\right)\cdot 101 + \left(28 a^{2} + 55 a + 43\right)\cdot 101^{2} + \left(35 a^{2} + 98 a + 54\right)\cdot 101^{3} + \left(41 a^{2} + 53 a + 55\right)\cdot 101^{4} + \left(7 a^{2} + 73 a + 94\right)\cdot 101^{5} + \left(75 a^{2} + 68 a + 85\right)\cdot 101^{6} + \left(100 a^{2} + 73 a + 31\right)\cdot 101^{7} + \left(18 a^{2} + 51 a + 91\right)\cdot 101^{8} + \left(59 a^{2} + 23 a + 43\right)\cdot 101^{9} + \left(75 a^{2} + 41 a + 1\right)\cdot 101^{10} + \left(83 a^{2} + 30 a + 59\right)\cdot 101^{11} + \left(51 a^{2} + 59 a + 83\right)\cdot 101^{12} +O\left(101^{ 13 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 7 a^{2} + 57 a + 41 + \left(91 a^{2} + 85 a + 64\right)\cdot 101 + \left(84 a^{2} + 19 a + 54\right)\cdot 101^{2} + \left(82 a^{2} + 57 a + 48\right)\cdot 101^{3} + \left(92 a^{2} + 69 a + 57\right)\cdot 101^{4} + \left(76 a^{2} + 34 a + 31\right)\cdot 101^{5} + \left(100 a^{2} + 13 a + 36\right)\cdot 101^{6} + \left(84 a^{2} + 25 a\right)\cdot 101^{7} + \left(54 a^{2} + 22 a + 62\right)\cdot 101^{8} + \left(82 a^{2} + 84 a + 90\right)\cdot 101^{9} + \left(36 a^{2} + 79 a + 24\right)\cdot 101^{10} + \left(98 a^{2} + 52 a + 88\right)\cdot 101^{11} + \left(20 a^{2} + 84 a + 21\right)\cdot 101^{12} +O\left(101^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,9,2)(3,8,4)(5,7,6)$ |
| $(2,5,4)(6,9,8)$ |
| $(2,8)(3,7)(4,9)(5,6)$ |
| $(1,7,3)(2,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,5)(2,7)(3,4)(8,9)$ |
$0$ |
| $2$ |
$3$ |
$(1,3,7)(2,4,5)(6,9,8)$ |
$-3$ |
| $3$ |
$3$ |
$(1,7,3)(2,4,5)$ |
$0$ |
| $3$ |
$3$ |
$(1,3,7)(2,5,4)$ |
$0$ |
| $6$ |
$3$ |
$(1,9,2)(3,8,4)(5,7,6)$ |
$0$ |
| $6$ |
$3$ |
$(1,2,9)(3,4,8)(5,6,7)$ |
$0$ |
| $6$ |
$3$ |
$(1,8,2)(3,6,4)(5,7,9)$ |
$0$ |
| $9$ |
$6$ |
$(1,4,7,5,3,2)(8,9)$ |
$0$ |
| $9$ |
$6$ |
$(1,2,3,5,7,4)(8,9)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
