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:
| $r_{ 1 }$ |
$=$ |
$ 2 a + 4 + \left(10 a + 21\right)\cdot 23 + 12\cdot 23^{2} + \left(14 a + 2\right)\cdot 23^{3} + 13 a\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 18\cdot 23 + 15\cdot 23^{2} + 7\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 + 8\cdot 23 + 6\cdot 23^{2} + 13\cdot 23^{3} + 9\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 a + 10 + \left(18 a + 11\right)\cdot 23 + \left(22 a + 1\right)\cdot 23^{2} + \left(4 a + 6\right)\cdot 23^{3} + \left(14 a + 19\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a + 8 + \left(12 a + 16\right)\cdot 23 + \left(22 a + 3\right)\cdot 23^{2} + \left(8 a + 7\right)\cdot 23^{3} + \left(9 a + 13\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 a + 5 + \left(4 a + 16\right)\cdot 23 + 5\cdot 23^{2} + \left(18 a + 16\right)\cdot 23^{3} + \left(8 a + 19\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4,6)$ |
| $(3,5)(4,6)$ |
| $(1,3,5)(2,4,6)$ |
| $(1,4)(2,3)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $3$ | $2$ | $(1,4)(2,3)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $9$ | $2$ | $(1,5)(2,6)$ | $0$ |
| $2$ | $3$ | $(1,3,5)(2,4,6)$ | $-2$ |
| $2$ | $3$ | $(1,3,5)(2,6,4)$ | $-2$ |
| $4$ | $3$ | $(2,4,6)$ | $1$ |
| $6$ | $6$ | $(1,6,3,2,5,4)$ | $0$ |
| $6$ | $6$ | $(1,4,5,6,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
