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 + 8 + \left(19 a + 1\right)\cdot 23 + \left(4 a + 4\right)\cdot 23^{2} + 19 a\cdot 23^{3} + \left(10 a + 18\right)\cdot 23^{4} + \left(4 a + 10\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a + 18 + \left(20 a + 12\right)\cdot 23 + \left(11 a + 5\right)\cdot 23^{2} + \left(4 a + 18\right)\cdot 23^{3} + \left(15 a + 6\right)\cdot 23^{4} + \left(4 a + 12\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 + 2\cdot 23 + 19\cdot 23^{2} + 14\cdot 23^{3} + 12\cdot 23^{4} + 19\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a + 6 + \left(3 a + 19\right)\cdot 23 + \left(18 a + 16\right)\cdot 23^{2} + \left(3 a + 10\right)\cdot 23^{3} + \left(12 a + 20\right)\cdot 23^{4} + \left(18 a + 8\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 + 3\cdot 23 + 14\cdot 23^{2} + 9\cdot 23^{3} + 23^{4} + 11\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 a + 20 + \left(2 a + 5\right)\cdot 23 + \left(11 a + 9\right)\cdot 23^{2} + \left(18 a + 15\right)\cdot 23^{3} + \left(7 a + 9\right)\cdot 23^{4} + \left(18 a + 6\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3)(4,5)$ |
| $(3,6)(4,5)$ |
| $(1,3)(2,5)(4,6)$ |
| $(1,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $3$ |
$2$ |
$(1,3)(2,5)(4,6)$ |
$0$ |
| $3$ |
$2$ |
$(1,6)(2,5)(3,4)$ |
$0$ |
| $9$ |
$2$ |
$(2,3)(4,5)$ |
$0$ |
| $2$ |
$3$ |
$(1,5,4)(2,3,6)$ |
$-2$ |
| $2$ |
$3$ |
$(1,4,5)(2,3,6)$ |
$-2$ |
| $4$ |
$3$ |
$(1,4,5)$ |
$1$ |
| $6$ |
$6$ |
$(1,3,5,6,4,2)$ |
$0$ |
| $6$ |
$6$ |
$(1,6,4,2,5,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
