Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{2} + 21 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 a + 17 + \left(19 a + 6\right)\cdot 23 + \left(17 a + 3\right)\cdot 23^{2} + \left(a + 7\right)\cdot 23^{3} + \left(22 a + 13\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a + 16 + \left(12 a + 2\right)\cdot 23 + \left(18 a + 22\right)\cdot 23^{2} + \left(22 a + 20\right)\cdot 23^{3} + 21 a\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 6\cdot 23 + 18\cdot 23^{2} + 11\cdot 23^{3} + 14\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 a + 6 + \left(3 a + 16\right)\cdot 23 + \left(5 a + 19\right)\cdot 23^{2} + \left(21 a + 15\right)\cdot 23^{3} + 9\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 a + 7 + \left(10 a + 20\right)\cdot 23 + 4 a\cdot 23^{2} + 2\cdot 23^{3} + \left(a + 22\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 3 + 16\cdot 23 + 4\cdot 23^{2} + 11\cdot 23^{3} + 8\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,6)$ |
| $(1,6,2)(3,5,4)$ |
| $(1,3,5)(2,4,6)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $3$ |
$2$ |
$(2,5)(3,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,4)(2,6)(3,5)$ |
$1$ |
| $8$ |
$3$ |
$(1,6,2)(3,5,4)$ |
$0$ |
| $6$ |
$4$ |
$(1,5,4,2)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
