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:
| $r_{ 1 }$ |
$=$ |
$ 16 + 5\cdot 23 + 4\cdot 23^{2} + 23^{3} + 15\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 a + 15 + \left(20 a + 4\right)\cdot 23 + \left(9 a + 10\right)\cdot 23^{2} + \left(4 a + 3\right)\cdot 23^{3} + \left(9 a + 5\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 a + 7 + \left(a + 2\right)\cdot 23 + \left(11 a + 12\right)\cdot 23^{2} + \left(11 a + 13\right)\cdot 23^{3} + \left(13 a + 18\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 a + 8 + \left(21 a + 16\right)\cdot 23 + \left(11 a + 9\right)\cdot 23^{2} + \left(11 a + 2\right)\cdot 23^{3} + \left(9 a + 11\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 18 a + 2 + \left(2 a + 17\right)\cdot 23 + \left(13 a + 9\right)\cdot 23^{2} + \left(18 a + 2\right)\cdot 23^{3} + \left(13 a + 19\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
