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