Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 109 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 109 }$: $ x^{2} + 108 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 100 a + 48 + \left(10 a + 95\right)\cdot 109 + \left(67 a + 54\right)\cdot 109^{2} + \left(36 a + 50\right)\cdot 109^{3} + \left(49 a + 7\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 7\cdot 109 + 52\cdot 109^{2} + 38\cdot 109^{3} + 81\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a + 39 + \left(98 a + 6\right)\cdot 109 + \left(41 a + 2\right)\cdot 109^{2} + \left(72 a + 20\right)\cdot 109^{3} + \left(59 a + 20\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 + 45\cdot 109 + 89\cdot 109^{2} + 100\cdot 109^{3} + 28\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 103 a + 48 + \left(12 a + 22\right)\cdot 109 + \left(20 a + 6\right)\cdot 109^{2} + \left(80 a + 83\right)\cdot 109^{3} + \left(77 a + 95\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 a + 42 + \left(96 a + 41\right)\cdot 109 + \left(88 a + 13\right)\cdot 109^{2} + \left(28 a + 34\right)\cdot 109^{3} + \left(31 a + 93\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(4,5,6)$ |
| $(1,4)(2,5)(3,6)$ |
| $(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-2$ |
| $6$ |
$2$ |
$(4,5)$ |
$0$ |
| $9$ |
$2$ |
$(1,2)(4,5)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$1$ |
| $4$ |
$3$ |
$(1,2,3)$ |
$-2$ |
| $18$ |
$4$ |
$(1,4,2,5)(3,6)$ |
$0$ |
| $12$ |
$6$ |
$(1,5,2,6,3,4)$ |
$1$ |
| $12$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
