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