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