Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 107 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 107 }$: $ x^{3} + 5 x + 105 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 45 + 101\cdot 107 + 80\cdot 107^{2} + 23\cdot 107^{3} + 98\cdot 107^{4} + 45\cdot 107^{5} + 54\cdot 107^{6} + 70\cdot 107^{7} + 79\cdot 107^{8} + 44\cdot 107^{9} +O\left(107^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 49 + 82\cdot 107 + 32\cdot 107^{2} + 56\cdot 107^{3} + 19\cdot 107^{4} + 30\cdot 107^{5} + 44\cdot 107^{6} + 56\cdot 107^{7} + 71\cdot 107^{8} + 73\cdot 107^{9} +O\left(107^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 79 + 88\cdot 107 + 53\cdot 107^{2} + 103\cdot 107^{3} + 13\cdot 107^{4} + 32\cdot 107^{5} + 98\cdot 107^{6} + 102\cdot 107^{7} + 97\cdot 107^{9} +O\left(107^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a^{2} + 23 a + 43 + \left(79 a^{2} + 81 a + 15\right)\cdot 107 + \left(66 a^{2} + 100 a + 104\right)\cdot 107^{2} + \left(42 a^{2} + 38 a + 25\right)\cdot 107^{3} + \left(97 a^{2} + a + 60\right)\cdot 107^{4} + \left(17 a^{2} + 50 a + 46\right)\cdot 107^{5} + \left(15 a^{2} + 34 a + 52\right)\cdot 107^{6} + \left(18 a^{2} + 29 a + 22\right)\cdot 107^{7} + \left(51 a^{2} + 70 a + 38\right)\cdot 107^{8} + \left(104 a^{2} + 11 a + 68\right)\cdot 107^{9} +O\left(107^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 37 a^{2} + 101 a + 75 + \left(75 a^{2} + 103 a + 38\right)\cdot 107 + \left(90 a^{2} + 91 a + 41\right)\cdot 107^{2} + \left(49 a^{2} + 90 a + 14\right)\cdot 107^{3} + \left(73 a^{2} + 4 a + 16\right)\cdot 107^{4} + \left(42 a^{2} + 63 a + 22\right)\cdot 107^{5} + \left(55 a^{2} + 72 a + 8\right)\cdot 107^{6} + \left(86 a^{2} + 23 a + 72\right)\cdot 107^{7} + \left(42 a^{2} + 3 a + 81\right)\cdot 107^{8} + \left(20 a^{2} + 20 a + 37\right)\cdot 107^{9} +O\left(107^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 60 a^{2} + 84 a + 49 + \left(96 a^{2} + 53 a + 15\right)\cdot 107 + \left(58 a^{2} + 20 a + 9\right)\cdot 107^{2} + \left(98 a^{2} + 76 a + 98\right)\cdot 107^{3} + \left(76 a^{2} + 43 a + 48\right)\cdot 107^{4} + \left(68 a^{2} + 78 a + 63\right)\cdot 107^{5} + \left(85 a^{2} + 6 a + 39\right)\cdot 107^{6} + \left(30 a^{2} + 44 a + 28\right)\cdot 107^{7} + \left(2 a^{2} + 15 a + 89\right)\cdot 107^{8} + \left(73 a^{2} + 38 a + 94\right)\cdot 107^{9} +O\left(107^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 64 a^{2} + 90 a + 58 + \left(59 a^{2} + 28 a + 57\right)\cdot 107 + \left(56 a^{2} + 21 a + 34\right)\cdot 107^{2} + \left(14 a^{2} + 84 a + 39\right)\cdot 107^{3} + \left(43 a^{2} + 100 a + 93\right)\cdot 107^{4} + \left(46 a^{2} + 100 a + 105\right)\cdot 107^{5} + \left(36 a^{2} + 106 a + 51\right)\cdot 107^{6} + \left(2 a^{2} + 53 a + 5\right)\cdot 107^{7} + \left(13 a^{2} + 33 a + 18\right)\cdot 107^{8} + \left(89 a^{2} + 75 a + 17\right)\cdot 107^{9} +O\left(107^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 74 a^{2} + 77 a + 60 + \left(55 a^{2} + 12 a + 57\right)\cdot 107 + \left(51 a^{2} + 60 a + 91\right)\cdot 107^{2} + \left(45 a^{2} + 13 a + 63\right)\cdot 107^{3} + \left(74 a^{2} + 86 a + 40\right)\cdot 107^{4} + \left(31 a^{2} + 96 a + 11\right)\cdot 107^{5} + \left(66 a^{2} + 21 a + 82\right)\cdot 107^{6} + \left(77 a^{2} + 2 a + 41\right)\cdot 107^{7} + \left(87 a^{2} + 33 a + 17\right)\cdot 107^{8} + \left(49 a^{2} + 74 a + 53\right)\cdot 107^{9} +O\left(107^{ 10 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 80 a^{2} + 53 a + 80 + \left(61 a^{2} + 40 a + 77\right)\cdot 107 + \left(103 a^{2} + 26 a + 86\right)\cdot 107^{2} + \left(69 a^{2} + 17 a + 2\right)\cdot 107^{3} + \left(62 a^{2} + 84 a + 37\right)\cdot 107^{4} + \left(6 a^{2} + 38 a + 70\right)\cdot 107^{5} + \left(62 a^{2} + 78 a + 103\right)\cdot 107^{6} + \left(105 a^{2} + 60 a + 27\right)\cdot 107^{7} + \left(16 a^{2} + 58 a + 31\right)\cdot 107^{8} + \left(91 a^{2} + 101 a + 48\right)\cdot 107^{9} +O\left(107^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2,3)(4,7,5)$ |
| $(1,2)(4,6)(5,9)(7,8)$ |
| $(4,7,5)(6,8,9)$ |
| $(1,9,5,2,6,7,3,8,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,2)(4,6)(5,9)(7,8)$ |
$0$ |
| $2$ |
$3$ |
$(1,2,3)(4,5,7)(6,8,9)$ |
$-3$ |
| $3$ |
$3$ |
$(1,2,3)(4,7,5)$ |
$0$ |
| $3$ |
$3$ |
$(1,3,2)(4,5,7)$ |
$0$ |
| $9$ |
$6$ |
$(1,3)(4,6,7,8,5,9)$ |
$0$ |
| $9$ |
$6$ |
$(1,3)(4,9,5,8,7,6)$ |
$0$ |
| $6$ |
$9$ |
$(1,9,5,2,6,7,3,8,4)$ |
$0$ |
| $6$ |
$9$ |
$(1,6,5,2,8,7,3,9,4)$ |
$0$ |
| $6$ |
$9$ |
$(1,5,8,3,4,6,2,7,9)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
