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