Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 107 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 107 }$: $ x^{3} + 5 x + 105 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 9\cdot 107 + 19\cdot 107^{2} + 26\cdot 107^{3} + 24\cdot 107^{4} + 16\cdot 107^{5} + 34\cdot 107^{6} + 92\cdot 107^{7} + 91\cdot 107^{8} + 34\cdot 107^{9} + 54\cdot 107^{10} + 10\cdot 107^{11} +O\left(107^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 53 + 5\cdot 107 + 18\cdot 107^{2} + 22\cdot 107^{3} + 104\cdot 107^{4} + 26\cdot 107^{5} + 14\cdot 107^{6} + 80\cdot 107^{7} + 27\cdot 107^{8} + 102\cdot 107^{9} + 5\cdot 107^{10} + 74\cdot 107^{11} +O\left(107^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 84 + 6\cdot 107 + 59\cdot 107^{2} + 52\cdot 107^{3} + 91\cdot 107^{4} + 27\cdot 107^{5} + 82\cdot 107^{6} + 38\cdot 107^{7} + 10\cdot 107^{8} + 41\cdot 107^{9} + 20\cdot 107^{10} + 93\cdot 107^{11} +O\left(107^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a^{2} + 16 a + 58 + \left(52 a^{2} + 66 a + 75\right)\cdot 107 + \left(100 a^{2} + 66 a + 64\right)\cdot 107^{2} + \left(91 a^{2} + 55 a + 86\right)\cdot 107^{3} + \left(94 a^{2} + 8 a\right)\cdot 107^{4} + \left(46 a^{2} + 15 a + 18\right)\cdot 107^{5} + \left(40 a^{2} + 55 a + 44\right)\cdot 107^{6} + \left(31 a^{2} + 78 a + 50\right)\cdot 107^{7} + \left(5 a^{2} + 49 a + 17\right)\cdot 107^{8} + \left(99 a^{2} + 58 a + 11\right)\cdot 107^{9} + \left(68 a^{2} + 76 a + 10\right)\cdot 107^{10} + \left(43 a^{2} + 86 a + 65\right)\cdot 107^{11} +O\left(107^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 47 a^{2} + 59 a + 23 + \left(8 a^{2} + 78 a + 37\right)\cdot 107 + \left(13 a^{2} + 44 a + 94\right)\cdot 107^{2} + \left(95 a^{2} + 25\right)\cdot 107^{3} + \left(a^{2} + 43 a + 83\right)\cdot 107^{4} + \left(10 a^{2} + 97 a + 37\right)\cdot 107^{5} + \left(96 a^{2} + 69 a + 51\right)\cdot 107^{6} + \left(24 a^{2} + 61 a + 64\right)\cdot 107^{7} + \left(16 a^{2} + a + 89\right)\cdot 107^{8} + \left(57 a^{2} + 87 a + 49\right)\cdot 107^{9} + \left(24 a^{2} + 62 a + 40\right)\cdot 107^{10} + \left(70 a^{2} + 65 a + 82\right)\cdot 107^{11} +O\left(107^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 54 a^{2} + 64 a + 52 + \left(12 a^{2} + 53 a + 25\right)\cdot 107 + \left(96 a^{2} + 17 a + 59\right)\cdot 107^{2} + \left(66 a^{2} + 66 a + 52\right)\cdot 107^{3} + \left(a^{2} + 9 a + 33\right)\cdot 107^{4} + \left(50 a^{2} + 27 a + 67\right)\cdot 107^{5} + \left(55 a^{2} + 33 a + 53\right)\cdot 107^{6} + \left(55 a^{2} + 28 a + 97\right)\cdot 107^{7} + \left(49 a^{2} + 67 a + 14\right)\cdot 107^{8} + \left(23 a^{2} + 24 a + 88\right)\cdot 107^{9} + \left(6 a^{2} + 74 a + 70\right)\cdot 107^{10} + \left(20 a^{2} + a + 16\right)\cdot 107^{11} +O\left(107^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 56 a^{2} + 32 a + 53 + \left(46 a^{2} + 69 a + 21\right)\cdot 107 + \left(100 a^{2} + 102 a + 100\right)\cdot 107^{2} + \left(26 a^{2} + 50 a + 83\right)\cdot 107^{3} + \left(10 a^{2} + 55 a + 39\right)\cdot 107^{4} + \left(50 a^{2} + 101 a + 64\right)\cdot 107^{5} + \left(77 a^{2} + 88 a + 60\right)\cdot 107^{6} + \left(50 a^{2} + 73 a + 43\right)\cdot 107^{7} + \left(85 a^{2} + 55 a + 106\right)\cdot 107^{8} + \left(57 a^{2} + 68 a + 51\right)\cdot 107^{9} + \left(13 a^{2} + 74 a + 39\right)\cdot 107^{10} + \left(100 a^{2} + 61 a + 39\right)\cdot 107^{11} +O\left(107^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 70 a^{2} + 75 a + 34 + \left(56 a^{2} + 60 a + 101\right)\cdot 107 + \left(50 a^{2} + 86 a + 49\right)\cdot 107^{2} + \left(42 a^{2} + 92 a + 42\right)\cdot 107^{3} + \left(68 a^{2} + 80 a + 6\right)\cdot 107^{4} + \left(59 a^{2} + 40 a + 28\right)\cdot 107^{5} + \left(14 a^{2} + 102 a + 24\right)\cdot 107^{6} + \left(80 a^{2} + 39 a + 1\right)\cdot 107^{7} + \left(64 a^{2} + 48 a + 30\right)\cdot 107^{8} + \left(35 a^{2} + 6 a + 57\right)\cdot 107^{9} + \left(28 a^{2} + 59 a + 37\right)\cdot 107^{10} + \left(75 a^{2} + 4 a + 22\right)\cdot 107^{11} +O\left(107^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 90 a^{2} + 75 a + 65 + \left(37 a^{2} + 99 a + 38\right)\cdot 107 + \left(67 a^{2} + 2 a + 70\right)\cdot 107^{2} + \left(104 a^{2} + 55 a + 35\right)\cdot 107^{3} + \left(36 a^{2} + 16 a + 44\right)\cdot 107^{4} + \left(104 a^{2} + 39 a + 34\right)\cdot 107^{5} + \left(36 a^{2} + 78 a + 63\right)\cdot 107^{6} + \left(78 a^{2} + 38 a + 66\right)\cdot 107^{7} + \left(99 a^{2} + 98 a + 39\right)\cdot 107^{8} + \left(47 a^{2} + 75 a + 98\right)\cdot 107^{9} + \left(72 a^{2} + 80 a + 41\right)\cdot 107^{10} + \left(11 a^{2} + 100 a + 24\right)\cdot 107^{11} +O\left(107^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(2,3)(4,8)(5,6)(7,9)$ |
| $(1,3,2)(4,7,5)$ |
| $(4,7,5)(6,8,9)$ |
| $(1,5,8)(2,4,6)(3,7,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,9)(2,6)(3,8)(4,5)$ |
$0$ |
| $2$ |
$3$ |
$(1,3,2)(4,5,7)(6,8,9)$ |
$-3$ |
| $3$ |
$3$ |
$(1,3,2)(4,7,5)$ |
$0$ |
| $3$ |
$3$ |
$(1,2,3)(4,5,7)$ |
$0$ |
| $6$ |
$3$ |
$(1,5,8)(2,4,6)(3,7,9)$ |
$0$ |
| $6$ |
$3$ |
$(1,8,5)(2,6,4)(3,9,7)$ |
$0$ |
| $6$ |
$3$ |
$(1,7,9)(2,5,8)(3,4,6)$ |
$0$ |
| $9$ |
$6$ |
$(1,6,3,9,2,8)(4,5)$ |
$0$ |
| $9$ |
$6$ |
$(1,8,2,9,3,6)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
