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