Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 107 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 107 }$: $ x^{2} + 103 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 89 a + 61 + \left(94 a + 58\right)\cdot 107 + \left(8 a + 42\right)\cdot 107^{2} + \left(10 a + 96\right)\cdot 107^{3} + \left(71 a + 73\right)\cdot 107^{4} + \left(98 a + 27\right)\cdot 107^{5} +O\left(107^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 a + 77 + \left(103 a + 54\right)\cdot 107 + \left(95 a + 75\right)\cdot 107^{2} + \left(72 a + 53\right)\cdot 107^{3} + \left(61 a + 48\right)\cdot 107^{4} + \left(96 a + 15\right)\cdot 107^{5} +O\left(107^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 75 a + 98 + \left(3 a + 7\right)\cdot 107 + \left(11 a + 35\right)\cdot 107^{2} + \left(34 a + 35\right)\cdot 107^{3} + \left(45 a + 8\right)\cdot 107^{4} + \left(10 a + 19\right)\cdot 107^{5} +O\left(107^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 103 + 87\cdot 107 + 70\cdot 107^{2} + 26\cdot 107^{3} + 24\cdot 107^{4} + 72\cdot 107^{5} +O\left(107^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 47 + 34\cdot 107 + 80\cdot 107^{2} + 56\cdot 107^{3} + 73\cdot 107^{5} +O\left(107^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 84 a + 20 + \left(23 a + 19\right)\cdot 107 + \left(15 a + 105\right)\cdot 107^{2} + \left(4 a + 14\right)\cdot 107^{3} + \left(34 a + 3\right)\cdot 107^{4} + \left(73 a + 19\right)\cdot 107^{5} +O\left(107^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 23 a + 35 + \left(83 a + 30\right)\cdot 107 + \left(91 a + 35\right)\cdot 107^{2} + \left(102 a + 16\right)\cdot 107^{3} + \left(72 a + 28\right)\cdot 107^{4} + \left(33 a + 64\right)\cdot 107^{5} +O\left(107^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 18 a + 96 + \left(12 a + 27\right)\cdot 107 + \left(98 a + 90\right)\cdot 107^{2} + \left(96 a + 20\right)\cdot 107^{3} + \left(35 a + 27\right)\cdot 107^{4} + \left(8 a + 30\right)\cdot 107^{5} +O\left(107^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,4)(2,5,3)$ |
| $(1,3)(2,4)(5,8)$ |
| $(1,6,3,7)(2,4,8,5)$ |
| $(1,3)(2,8)(4,5)(6,7)$ |
| $(1,8,3,2)(4,7,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,3)(2,8)(4,5)(6,7)$ |
$-4$ |
| $12$ |
$2$ |
$(1,3)(2,4)(5,8)$ |
$0$ |
| $8$ |
$3$ |
$(1,8,4)(2,5,3)$ |
$1$ |
| $6$ |
$4$ |
$(1,8,3,2)(4,7,5,6)$ |
$0$ |
| $8$ |
$6$ |
$(1,3)(2,6,4,8,7,5)$ |
$-1$ |
| $6$ |
$8$ |
$(1,5,6,2,3,4,7,8)$ |
$0$ |
| $6$ |
$8$ |
$(1,4,6,8,3,5,7,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
