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