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:
| $r_{ 1 }$ |
$=$ |
$ 5 a + 10 + \left(13 a + 21\right)\cdot 23 + \left(13 a + 14\right)\cdot 23^{2} + \left(5 a + 2\right)\cdot 23^{3} + \left(4 a + 10\right)\cdot 23^{4} + \left(7 a + 6\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 a + 11 + 17 a\cdot 23 + 17 a\cdot 23^{2} + \left(16 a + 13\right)\cdot 23^{3} + 21 a\cdot 23^{4} + \left(17 a + 21\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 a + 22 + \left(19 a + 7\right)\cdot 23 + \left(a + 9\right)\cdot 23^{2} + \left(13 a + 16\right)\cdot 23^{3} + \left(19 a + 20\right)\cdot 23^{4} + \left(17 a + 22\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 a + 20 + \left(9 a + 19\right)\cdot 23 + \left(9 a + 5\right)\cdot 23^{2} + 17 a\cdot 23^{3} + \left(18 a + 13\right)\cdot 23^{4} + \left(15 a + 16\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 a + 5 + \left(3 a + 20\right)\cdot 23 + \left(21 a + 16\right)\cdot 23^{2} + \left(9 a + 17\right)\cdot 23^{3} + 3 a\cdot 23^{4} + \left(5 a + 16\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 3 a + 5 + \left(5 a + 16\right)\cdot 23 + \left(5 a + 17\right)\cdot 23^{2} + \left(6 a + 5\right)\cdot 23^{3} + \left(a + 4\right)\cdot 23^{4} + \left(5 a + 12\right)\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 7 + 15\cdot 23 + 16\cdot 23^{2} + 19\cdot 23^{3} + 20\cdot 23^{4} + 5\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 14 + 13\cdot 23 + 10\cdot 23^{2} + 16\cdot 23^{3} + 21\cdot 23^{4} + 13\cdot 23^{5} +O\left(23^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,6,8)(2,5,4,3)$ |
| $(1,5,6,3)(2,8,4,7)$ |
| $(1,3)(5,6)(7,8)$ |
| $(1,5,8)(3,7,6)$ |
| $(1,6)(2,4)(3,5)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,4)(3,5)(7,8)$ |
$-2$ |
$-2$ |
| $12$ |
$2$ |
$(1,3)(5,6)(7,8)$ |
$0$ |
$0$ |
| $8$ |
$3$ |
$(1,3,2)(4,6,5)$ |
$-1$ |
$-1$ |
| $6$ |
$4$ |
$(1,7,6,8)(2,5,4,3)$ |
$0$ |
$0$ |
| $8$ |
$6$ |
$(1,4,3,6,2,5)(7,8)$ |
$1$ |
$1$ |
| $6$ |
$8$ |
$(1,8,3,2,6,7,5,4)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ |
$8$ |
$(1,7,3,4,6,8,5,2)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
