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