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