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:
| $r_{ 1 }$ |
$=$ |
$ 7 a + 5 + \left(15 a + 16\right)\cdot 23 + \left(13 a + 17\right)\cdot 23^{2} + \left(4 a + 15\right)\cdot 23^{3} + 6 a\cdot 23^{4} + 11 a\cdot 23^{5} + \left(15 a + 3\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 12\cdot 23 + 8\cdot 23^{2} + 10\cdot 23^{3} + 12\cdot 23^{4} + 7\cdot 23^{5} + 20\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 2 a + 6 + \left(9 a + 19\right)\cdot 23 + \left(11 a + 18\right)\cdot 23^{2} + 15 a\cdot 23^{3} + \left(15 a + 21\right)\cdot 23^{4} + \left(11 a + 13\right)\cdot 23^{5} + \left(5 a + 7\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 a + 19 + \left(7 a + 16\right)\cdot 23 + \left(9 a + 6\right)\cdot 23^{2} + \left(18 a + 11\right)\cdot 23^{3} + \left(16 a + 8\right)\cdot 23^{4} + \left(11 a + 16\right)\cdot 23^{5} + \left(7 a + 22\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 18 + 14\cdot 23 + 7\cdot 23^{2} + 10\cdot 23^{3} + 12\cdot 23^{4} + 9\cdot 23^{5} + 8\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 a + 10 + \left(13 a + 12\right)\cdot 23 + \left(11 a + 9\right)\cdot 23^{2} + \left(7 a + 20\right)\cdot 23^{3} + \left(7 a + 13\right)\cdot 23^{4} + \left(11 a + 21\right)\cdot 23^{5} + \left(17 a + 6\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6,3)(2,5,4)$ |
| $(1,3)(2,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,6)(3,5)$ | $-1$ |
| $20$ | $3$ | $(1,6,3)(2,5,4)$ | $0$ |
| $12$ | $5$ | $(2,4,5,6,3)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $12$ | $5$ | $(2,5,3,4,6)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
