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:
| $r_{ 1 }$ |
$=$ |
$ 4 + 18\cdot 23 + 20\cdot 23^{2} + 5\cdot 23^{3} + 15\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 + 14\cdot 23 + 22\cdot 23^{2} + 5\cdot 23^{3} + 11\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 a + 19 + \left(22 a + 7\right)\cdot 23 + \left(21 a + 12\right)\cdot 23^{2} + \left(21 a + 20\right)\cdot 23^{3} + \left(17 a + 21\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 14 a + 19 + \left(11 a + 20\right)\cdot 23 + \left(6 a + 11\right)\cdot 23^{2} + \left(22 a + 12\right)\cdot 23^{3} + \left(3 a + 22\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a + 1 + \left(11 a + 7\right)\cdot 23 + \left(16 a + 13\right)\cdot 23^{2} + 4\cdot 23^{3} + \left(19 a + 8\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 a + 8 + \left(a + 11\right)\cdot 23^{2} + \left(a + 19\right)\cdot 23^{3} + \left(5 a + 12\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(1,4)$ |
| $(1,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(2,3)$ | $0$ |
| $9$ | $2$ | $(1,4)(2,3)$ | $0$ |
| $4$ | $3$ | $(1,4,5)(2,3,6)$ | $1$ |
| $4$ | $3$ | $(2,3,6)$ | $-2$ |
| $18$ | $4$ | $(1,2,4,3)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,2,4,3,5,6)$ | $-1$ |
| $12$ | $6$ | $(1,4,5)(2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
