Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{2} + 21 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 a + 13 + \left(12 a + 19\right)\cdot 23 + \left(19 a + 2\right)\cdot 23^{2} + \left(19 a + 15\right)\cdot 23^{3} + \left(13 a + 6\right)\cdot 23^{4} + \left(16 a + 9\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a + 11 + 10 a\cdot 23 + \left(3 a + 6\right)\cdot 23^{2} + \left(3 a + 12\right)\cdot 23^{3} + \left(9 a + 14\right)\cdot 23^{4} + \left(6 a + 5\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 a + 17 + 11\cdot 23 + \left(22 a + 17\right)\cdot 23^{2} + \left(14 a + 1\right)\cdot 23^{3} + \left(14 a + 13\right)\cdot 23^{4} + \left(5 a + 8\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 + 17\cdot 23 + 4\cdot 23^{4} + 11\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 a + 9 + \left(22 a + 17\right)\cdot 23 + 14\cdot 23^{2} + \left(8 a + 9\right)\cdot 23^{3} + \left(8 a + 4\right)\cdot 23^{4} + \left(17 a + 5\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 13 + 20\cdot 23 + 15\cdot 23^{2} + 22\cdot 23^{3} + 9\cdot 23^{4} + 16\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 6 a + \left(11 a + 17\right)\cdot 23 + \left(11 a + 22\right)\cdot 23^{2} + \left(12 a + 19\right)\cdot 23^{3} + \left(19 a + 17\right)\cdot 23^{4} + \left(2 a + 1\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 17 a + 12 + \left(11 a + 10\right)\cdot 23 + \left(11 a + 11\right)\cdot 23^{2} + \left(10 a + 10\right)\cdot 23^{3} + \left(3 a + 21\right)\cdot 23^{4} + \left(20 a + 10\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,3)(4,6)(5,8)$ |
| $(2,6,5)(3,8,4)$ |
| $(1,7)(2,8)(3,5)(4,6)$ |
| $(1,6,7,4)(2,3,8,5)$ |
| $(1,2,7,8)(3,6,5,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,7)(2,8)(3,5)(4,6)$ |
$-4$ |
| $12$ |
$2$ |
$(2,3)(4,6)(5,8)$ |
$0$ |
| $8$ |
$3$ |
$(1,5,8)(2,7,3)$ |
$1$ |
| $6$ |
$4$ |
$(1,2,7,8)(3,6,5,4)$ |
$0$ |
| $8$ |
$6$ |
$(1,2,5,7,8,3)(4,6)$ |
$-1$ |
| $6$ |
$8$ |
$(1,3,4,2,7,5,6,8)$ |
$0$ |
| $6$ |
$8$ |
$(1,5,4,8,7,3,6,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
