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:
| $r_{ 1 }$ |
$=$ |
$ 6 a + 6 + \left(10 a + 11\right)\cdot 23 + \left(8 a + 14\right)\cdot 23^{2} + \left(4 a + 9\right)\cdot 23^{3} + \left(9 a + 12\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 a + 20 + \left(14 a + 9\right)\cdot 23 + \left(10 a + 11\right)\cdot 23^{2} + \left(14 a + 7\right)\cdot 23^{3} + 7\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 + 2\cdot 23 + 17\cdot 23^{2} + 10\cdot 23^{3} + 6\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 a + 1 + \left(8 a + 13\right)\cdot 23 + \left(12 a + 18\right)\cdot 23^{2} + \left(8 a + 2\right)\cdot 23^{3} + \left(22 a + 17\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 + 6\cdot 23 + 9\cdot 23^{2} + 5\cdot 23^{3} + 22\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 a + 18 + \left(12 a + 2\right)\cdot 23 + \left(14 a + 21\right)\cdot 23^{2} + \left(18 a + 9\right)\cdot 23^{3} + \left(13 a + 3\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)(2,5)(3,4)$ |
| $(1,4,6,5,2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,6)(2,5)(3,4)$ | $-2$ |
| $15$ | $2$ | $(2,3)(5,6)$ | $0$ |
| $20$ | $3$ | $(1,6,2)(3,4,5)$ | $1$ |
| $30$ | $4$ | $(2,6,3,5)$ | $0$ |
| $24$ | $5$ | $(1,5,3,6,4)$ | $-1$ |
| $20$ | $6$ | $(1,4,6,5,2,3)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
