Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 107 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 107 }$: $ x^{3} + 5 x + 105 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 35 + 19\cdot 107 + 8\cdot 107^{2} + 80\cdot 107^{3} + 45\cdot 107^{5} + 22\cdot 107^{6} + 87\cdot 107^{7} + 69\cdot 107^{8} + 107^{9} + 69\cdot 107^{10} +O\left(107^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 68 + 98\cdot 107 + 28\cdot 107^{2} + 76\cdot 107^{3} + 104\cdot 107^{4} + 97\cdot 107^{5} + 34\cdot 107^{6} + 39\cdot 107^{7} + 27\cdot 107^{8} + 73\cdot 107^{9} + 105\cdot 107^{10} +O\left(107^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 70 + 30\cdot 107 + 44\cdot 107^{2} + 22\cdot 107^{3} + 44\cdot 107^{4} + 102\cdot 107^{5} + 73\cdot 107^{6} + 15\cdot 107^{7} + 91\cdot 107^{8} + 5\cdot 107^{9} + 14\cdot 107^{10} +O\left(107^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 a^{2} + 41 a + 62 + \left(71 a^{2} + 46 a + 19\right)\cdot 107 + \left(106 a^{2} + 96 a + 64\right)\cdot 107^{2} + \left(100 a^{2} + 93 a + 65\right)\cdot 107^{3} + \left(46 a^{2} + 96 a + 75\right)\cdot 107^{4} + \left(98 a^{2} + 43 a + 105\right)\cdot 107^{5} + \left(82 a^{2} + 12 a + 74\right)\cdot 107^{6} + \left(56 a^{2} + 59 a + 65\right)\cdot 107^{7} + \left(37 a^{2} + 81 a + 2\right)\cdot 107^{8} + \left(72 a^{2} + 30 a + 91\right)\cdot 107^{9} + \left(63 a^{2} + 101 a + 11\right)\cdot 107^{10} +O\left(107^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 44 a^{2} + 28 a + 104 + \left(105 a^{2} + 93 a + 26\right)\cdot 107 + \left(57 a^{2} + 47 a + 80\right)\cdot 107^{2} + \left(55 a^{2} + 25 a + 56\right)\cdot 107^{3} + \left(70 a^{2} + 83 a + 11\right)\cdot 107^{4} + \left(51 a^{2} + 104 a + 21\right)\cdot 107^{5} + \left(47 a^{2} + 34 a + 28\right)\cdot 107^{6} + \left(98 a^{2} + 57 a + 26\right)\cdot 107^{7} + \left(87 a^{2} + 22 a + 99\right)\cdot 107^{8} + \left(67 a^{2} + 32 a + 75\right)\cdot 107^{9} + \left(13 a^{2} + 94 a + 94\right)\cdot 107^{10} +O\left(107^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 45 a^{2} + 34 a + 100 + \left(a^{2} + 96 a + 29\right)\cdot 107 + \left(2 a^{2} + 16 a + 21\right)\cdot 107^{2} + \left(103 a^{2} + 84 a + 91\right)\cdot 107^{3} + \left(24 a^{2} + 11 a + 42\right)\cdot 107^{4} + \left(45 a^{2} + 16 a + 77\right)\cdot 107^{5} + \left(38 a^{2} + 28 a + 71\right)\cdot 107^{6} + \left(18 a^{2} + 102 a + 101\right)\cdot 107^{7} + \left(7 a^{2} + 101 a + 47\right)\cdot 107^{8} + \left(57 a^{2} + 76 a + 99\right)\cdot 107^{9} + \left(88 a^{2} + 68 a + 75\right)\cdot 107^{10} +O\left(107^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 53 a^{2} + 38 a + 27 + \left(37 a^{2} + 74 a + 50\right)\cdot 107 + \left(49 a^{2} + 69 a + 51\right)\cdot 107^{2} + \left(57 a^{2} + 94 a + 27\right)\cdot 107^{3} + \left(96 a^{2} + 33 a + 98\right)\cdot 107^{4} + \left(63 a^{2} + 65 a + 61\right)\cdot 107^{5} + \left(83 a^{2} + 59 a + 41\right)\cdot 107^{6} + \left(58 a^{2} + 97 a + 72\right)\cdot 107^{7} + \left(88 a^{2} + 2 a + 65\right)\cdot 107^{8} + \left(73 a^{2} + 44 a + 24\right)\cdot 107^{9} + \left(29 a^{2} + 18 a + 41\right)\cdot 107^{10} +O\left(107^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 64 a^{2} + 72 a + 92 + \left(15 a^{2} + 55 a + 5\right)\cdot 107 + \left(25 a^{2} + 16 a + 27\right)\cdot 107^{2} + \left(36 a^{2} + 11 a + 11\right)\cdot 107^{3} + \left(11 a^{2} + 93 a + 33\right)\cdot 107^{4} + \left(98 a^{2} + 23 a + 75\right)\cdot 107^{5} + \left(46 a^{2} + 13 a + 28\right)\cdot 107^{6} + \left(94 a^{2} + 84 a + 34\right)\cdot 107^{7} + \left(49 a^{2} + 18 a + 83\right)\cdot 107^{8} + \left(4 a^{2} + 54 a + 66\right)\cdot 107^{9} + \left(99 a^{2} + 62 a + 39\right)\cdot 107^{10} +O\left(107^{ 11 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 105 a^{2} + a + 86 + \left(89 a^{2} + 62 a + 39\right)\cdot 107 + \left(79 a^{2} + 73 a + 102\right)\cdot 107^{2} + \left(74 a^{2} + 11 a + 103\right)\cdot 107^{3} + \left(70 a^{2} + 2 a + 16\right)\cdot 107^{4} + \left(70 a^{2} + 67 a + 55\right)\cdot 107^{5} + \left(21 a^{2} + 65 a + 51\right)\cdot 107^{6} + \left(101 a^{2} + 27 a + 92\right)\cdot 107^{7} + \left(49 a^{2} + 93 a + 47\right)\cdot 107^{8} + \left(45 a^{2} + 82 a + 96\right)\cdot 107^{9} + \left(26 a^{2} + 82 a + 82\right)\cdot 107^{10} +O\left(107^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(4,6)(5,8)(7,9)$ |
| $(4,7,5)(6,8,9)$ |
| $(2,3)(4,5)(6,8)$ |
| $(1,3,2)(4,5,7)(6,8,9)$ |
| $(1,5,8)(2,4,6)(3,7,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(2,3)(5,7)(6,9)$ |
$0$ |
| $9$ |
$2$ |
$(4,6)(5,8)(7,9)$ |
$-2$ |
| $9$ |
$2$ |
$(1,7)(2,4)(3,5)(6,8)$ |
$0$ |
| $2$ |
$3$ |
$(1,3,2)(4,5,7)(6,8,9)$ |
$-3$ |
| $6$ |
$3$ |
$(1,8,4)(2,6,7)(3,9,5)$ |
$0$ |
| $6$ |
$3$ |
$(1,3,2)(4,7,5)$ |
$0$ |
| $12$ |
$3$ |
$(1,5,8)(2,4,6)(3,7,9)$ |
$0$ |
| $18$ |
$6$ |
$(1,4,8)(2,5,6,3,7,9)$ |
$0$ |
| $18$ |
$6$ |
$(1,3,2)(4,8,7,6,5,9)$ |
$1$ |
| $18$ |
$6$ |
$(1,4,3,7,2,5)(6,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
