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:
| $r_{ 1 }$ |
$=$ |
$ 6\cdot 23 + 15\cdot 23^{2} + 3\cdot 23^{3} + 18\cdot 23^{4} + 15\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 a + 11 + \left(16 a + 11\right)\cdot 23 + \left(4 a + 12\right)\cdot 23^{2} + \left(22 a + 16\right)\cdot 23^{3} + \left(3 a + 18\right)\cdot 23^{4} + \left(21 a + 18\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 + 8\cdot 23 + 16\cdot 23^{2} + 21\cdot 23^{3} + 5\cdot 23^{4} + 8\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a + 3 + \left(6 a + 4\right)\cdot 23 + \left(18 a + 6\right)\cdot 23^{2} + 5\cdot 23^{3} + \left(19 a + 15\right)\cdot 23^{4} + \left(a + 3\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a + 7 + \left(16 a + 14\right)\cdot 23 + \left(4 a + 13\right)\cdot 23^{2} + \left(22 a + 11\right)\cdot 23^{3} + \left(3 a + 6\right)\cdot 23^{4} + \left(21 a + 11\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 2 a + 7 + \left(6 a + 1\right)\cdot 23 + \left(18 a + 5\right)\cdot 23^{2} + 10\cdot 23^{3} + \left(19 a + 4\right)\cdot 23^{4} + \left(a + 11\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,6)(4,5)$ |
| $(1,3)(2,5)(4,6)$ |
| $(1,2,6)(3,5,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,5)(4,6)$ |
$-2$ |
| $3$ |
$2$ |
$(2,6)(4,5)$ |
$0$ |
| $3$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,2,6)(3,5,4)$ |
$-1$ |
| $2$ |
$6$ |
$(1,5,6,3,2,4)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
