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