Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{2} + 21 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 12\cdot 23 + 16\cdot 23^{2} + 23^{3} + 3\cdot 23^{4} + 3\cdot 23^{5} + 16\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 a + 7 + \left(16 a + 21\right)\cdot 23 + \left(11 a + 16\right)\cdot 23^{2} + \left(7 a + 16\right)\cdot 23^{3} + \left(16 a + 11\right)\cdot 23^{4} + \left(19 a + 11\right)\cdot 23^{5} + \left(18 a + 2\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 a + 6 + \left(6 a + 20\right)\cdot 23 + 11 a\cdot 23^{2} + \left(15 a + 20\right)\cdot 23^{3} + \left(6 a + 13\right)\cdot 23^{4} + \left(3 a + 11\right)\cdot 23^{5} + \left(4 a + 20\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 a + \left(9 a + 4\right)\cdot 23 + \left(a + 9\right)\cdot 23^{2} + \left(8 a + 19\right)\cdot 23^{3} + \left(8 a + 15\right)\cdot 23^{4} + 13 a\cdot 23^{5} + 21\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 a + 1 + \left(13 a + 11\right)\cdot 23 + \left(21 a + 2\right)\cdot 23^{2} + \left(14 a + 11\right)\cdot 23^{3} + \left(14 a + 1\right)\cdot 23^{4} + \left(9 a + 19\right)\cdot 23^{5} + \left(22 a + 8\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)(3,4)$ |
| $(1,5,2,3,4)$ |
| $(1,3,2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $5$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $5$ | $4$ | $(1,3,2,4)$ | $0$ |
| $5$ | $4$ | $(1,4,2,3)$ | $0$ |
| $4$ | $5$ | $(1,5,2,3,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
