Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 28 + 29\cdot 43 + 15\cdot 43^{2} + 16\cdot 43^{3} + 4\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 a + 27 + \left(29 a + 8\right)\cdot 43 + \left(33 a + 27\right)\cdot 43^{2} + \left(21 a + 13\right)\cdot 43^{3} + \left(38 a + 2\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 41 a + 9 + \left(39 a + 7\right)\cdot 43 + \left(37 a + 36\right)\cdot 43^{2} + \left(25 a + 40\right)\cdot 43^{3} + \left(33 a + 36\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 41 a + 29 + \left(13 a + 35\right)\cdot 43 + \left(9 a + 31\right)\cdot 43^{2} + \left(21 a + 1\right)\cdot 43^{3} + \left(4 a + 19\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 + 41\cdot 43 + 26\cdot 43^{2} + 27\cdot 43^{3} + 21\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 2 a + 7 + \left(3 a + 6\right)\cdot 43 + \left(5 a + 34\right)\cdot 43^{2} + \left(17 a + 28\right)\cdot 43^{3} + \left(9 a + 1\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(1,3)$ |
| $(1,3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $6$ |
$2$ |
$(3,6)$ |
$2$ |
| $9$ |
$2$ |
$(3,6)(4,5)$ |
$0$ |
| $4$ |
$3$ |
$(1,3,6)$ |
$1$ |
| $4$ |
$3$ |
$(1,3,6)(2,4,5)$ |
$-2$ |
| $18$ |
$4$ |
$(1,2)(3,5,6,4)$ |
$0$ |
| $12$ |
$6$ |
$(1,4,3,5,6,2)$ |
$0$ |
| $12$ |
$6$ |
$(2,4,5)(3,6)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
