Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{2} + 21 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 a + 16 + \left(19 a + 17\right)\cdot 23 + \left(a + 19\right)\cdot 23^{2} + \left(13 a + 5\right)\cdot 23^{3} + 12\cdot 23^{4} + \left(9 a + 1\right)\cdot 23^{5} + \left(11 a + 17\right)\cdot 23^{6} + \left(18 a + 18\right)\cdot 23^{7} +O\left(23^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 16\cdot 23 + 15\cdot 23^{2} + 15\cdot 23^{3} + 3\cdot 23^{4} + 14\cdot 23^{5} + 19\cdot 23^{6} + 23^{7} +O\left(23^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 6 + \left(12 a + 8\right)\cdot 23 + \left(11 a + 7\right)\cdot 23^{2} + \left(6 a + 14\right)\cdot 23^{3} + \left(13 a + 7\right)\cdot 23^{4} + \left(5 a + 5\right)\cdot 23^{5} + 18 a\cdot 23^{6} + \left(19 a + 10\right)\cdot 23^{7} +O\left(23^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 a + 22 + \left(3 a + 6\right)\cdot 23 + \left(21 a + 4\right)\cdot 23^{2} + \left(9 a + 7\right)\cdot 23^{3} + 22 a\cdot 23^{4} + \left(13 a + 19\right)\cdot 23^{5} + \left(11 a + 7\right)\cdot 23^{6} + \left(4 a + 21\right)\cdot 23^{7} +O\left(23^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 + 16\cdot 23 + 3\cdot 23^{2} + 10\cdot 23^{3} + 17\cdot 23^{4} + 2\cdot 23^{5} + 16\cdot 23^{6} + 8\cdot 23^{7} +O\left(23^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 16 a + 20 + \left(10 a + 2\right)\cdot 23 + \left(11 a + 18\right)\cdot 23^{2} + \left(16 a + 15\right)\cdot 23^{3} + \left(9 a + 4\right)\cdot 23^{4} + \left(17 a + 3\right)\cdot 23^{5} + \left(4 a + 8\right)\cdot 23^{6} + \left(3 a + 8\right)\cdot 23^{7} +O\left(23^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$8$ |
$8$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
$0$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
$-1$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$-1$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
$0$ |
| $72$ |
$5$ |
$(1,2,3,4,5)$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $72$ |
$5$ |
$(1,3,4,5,2)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
