Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 a + 34 + \left(29 a + 26\right)\cdot 43 + \left(5 a + 32\right)\cdot 43^{2} + \left(30 a + 2\right)\cdot 43^{3} + \left(31 a + 31\right)\cdot 43^{4} + \left(17 a + 2\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 + 12\cdot 43 + 36\cdot 43^{2} + 36\cdot 43^{3} + 30\cdot 43^{4} + 30\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a + 40 + \left(29 a + 8\right)\cdot 43 + \left(5 a + 9\right)\cdot 43^{2} + \left(30 a + 1\right)\cdot 43^{3} + \left(31 a + 11\right)\cdot 43^{4} + \left(17 a + 2\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 a + 1 + \left(13 a + 34\right)\cdot 43 + \left(37 a + 28\right)\cdot 43^{2} + \left(12 a + 25\right)\cdot 43^{3} + \left(11 a + 12\right)\cdot 43^{4} + \left(25 a + 31\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 + 37\cdot 43 + 12\cdot 43^{2} + 35\cdot 43^{3} + 10\cdot 43^{4} + 30\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 a + 38 + \left(13 a + 8\right)\cdot 43 + \left(37 a + 9\right)\cdot 43^{2} + \left(12 a + 27\right)\cdot 43^{3} + \left(11 a + 32\right)\cdot 43^{4} + \left(25 a + 31\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,6)(4,5)$ |
| $(1,3)(2,5)(4,6)$ |
| $(1,2,6)(3,5,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,5)(4,6)$ |
$-2$ |
| $3$ |
$2$ |
$(2,6)(4,5)$ |
$0$ |
| $3$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,2,6)(3,5,4)$ |
$-1$ |
| $2$ |
$6$ |
$(1,5,6,3,2,4)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
