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:
| $r_{ 1 }$ |
$=$ |
$ 21 a + 6 + \left(12 a + 7\right)\cdot 23 + \left(7 a + 1\right)\cdot 23^{2} + \left(22 a + 12\right)\cdot 23^{3} + \left(17 a + 2\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 4\cdot 23 + 10\cdot 23^{2} + 22\cdot 23^{3} + 22\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 2 a + 2 + \left(10 a + 12\right)\cdot 23 + \left(15 a + 3\right)\cdot 23^{2} + 3\cdot 23^{3} + \left(5 a + 16\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 + 3\cdot 23 + 18\cdot 23^{2} + 7\cdot 23^{3} + 4\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 a + 18 + \left(8 a + 19\right)\cdot 23 + \left(7 a + 14\right)\cdot 23^{2} + \left(17 a + 9\right)\cdot 23^{3} + 19\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 8 a + 2 + \left(14 a + 22\right)\cdot 23 + \left(15 a + 20\right)\cdot 23^{2} + \left(5 a + 13\right)\cdot 23^{3} + \left(22 a + 3\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(2,5)$ |
| $(2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(1,3)$ | $2$ |
| $9$ | $2$ | $(1,3)(2,5)$ | $0$ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $-2$ |
| $4$ | $3$ | $(1,3,4)$ | $1$ |
| $18$ | $4$ | $(1,5,3,2)(4,6)$ | $0$ |
| $12$ | $6$ | $(1,5,3,6,4,2)$ | $0$ |
| $12$ | $6$ | $(1,3)(2,5,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
