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:
| $r_{ 1 }$ |
$=$ |
$ 12 a + 2 + 22\cdot 23 + \left(a + 1\right)\cdot 23^{2} + \left(9 a + 16\right)\cdot 23^{3} + 15\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 a + 3 + \left(22 a + 11\right)\cdot 23 + \left(21 a + 3\right)\cdot 23^{2} + \left(13 a + 10\right)\cdot 23^{3} + \left(22 a + 7\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 + 2\cdot 23 + 19\cdot 23^{2} + 4\cdot 23^{3} + 9\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 14 a + 3 + \left(20 a + 17\right)\cdot 23 + \left(16 a + 4\right)\cdot 23^{2} + \left(2 a + 11\right)\cdot 23^{3} + \left(14 a + 4\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 + 17\cdot 23 + 21\cdot 23^{2} + 3\cdot 23^{3} + 2\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a + 8 + \left(2 a + 21\right)\cdot 23 + \left(6 a + 17\right)\cdot 23^{2} + \left(20 a + 22\right)\cdot 23^{3} + \left(8 a + 6\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,6)(2,3,4)$ |
| $(1,3,4)(2,5,6)$ |
| $(3,5)(4,6)$ |
| $(1,6,2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $3$ |
$2$ |
$(1,2)(4,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,6)(2,4)(3,5)$ |
$-1$ |
| $8$ |
$3$ |
$(1,5,6)(2,3,4)$ |
$0$ |
| $6$ |
$4$ |
$(1,6,2,4)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
