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:
| $r_{ 1 }$ |
$=$ |
$ 15 a + 8 + \left(8 a + 17\right)\cdot 23 + \left(14 a + 16\right)\cdot 23^{2} + \left(13 a + 5\right)\cdot 23^{3} + \left(17 a + 15\right)\cdot 23^{4} + \left(3 a + 16\right)\cdot 23^{5} + \left(8 a + 2\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 a + 13 + \left(21 a + 19\right)\cdot 23 + \left(9 a + 11\right)\cdot 23^{2} + \left(20 a + 11\right)\cdot 23^{3} + \left(17 a + 2\right)\cdot 23^{4} + \left(9 a + 15\right)\cdot 23^{5} + \left(12 a + 14\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 21\cdot 23 + 15\cdot 23^{2} + 13\cdot 23^{3} + 19\cdot 23^{4} + 13\cdot 23^{5} + 6\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 a + 15 + \left(14 a + 19\right)\cdot 23 + \left(8 a + 13\right)\cdot 23^{2} + \left(9 a + 18\right)\cdot 23^{3} + \left(5 a + 13\right)\cdot 23^{4} + \left(19 a + 6\right)\cdot 23^{5} + \left(14 a + 15\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a + 17 + \left(a + 13\right)\cdot 23 + \left(13 a + 10\right)\cdot 23^{2} + \left(2 a + 19\right)\cdot 23^{3} + \left(5 a + 17\right)\cdot 23^{4} + \left(13 a + 16\right)\cdot 23^{5} + \left(10 a + 6\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,5)(3,4)$ |
| $(1,4,5,3)$ |
| $(1,4,2,3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $5$ | $2$ | $(1,5)(3,4)$ | $0$ |
| $5$ | $4$ | $(1,4,5,3)$ | $0$ |
| $5$ | $4$ | $(1,3,5,4)$ | $0$ |
| $4$ | $5$ | $(1,4,2,3,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
