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:
| $r_{ 1 }$ |
$=$ |
$ 10 + 22\cdot 43 + 29\cdot 43^{2} + 42\cdot 43^{3} + 27\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 a + 27 + \left(20 a + 2\right)\cdot 43 + \left(17 a + 15\right)\cdot 43^{2} + \left(2 a + 31\right)\cdot 43^{3} + \left(40 a + 7\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 a + 12 + \left(22 a + 38\right)\cdot 43 + \left(25 a + 11\right)\cdot 43^{2} + \left(40 a + 16\right)\cdot 43^{3} + \left(2 a + 2\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 a + 36 + \left(36 a + 18\right)\cdot 43 + \left(31 a + 36\right)\cdot 43^{2} + \left(37 a + 29\right)\cdot 43^{3} + \left(15 a + 12\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 a + 16 + \left(6 a + 32\right)\cdot 43 + \left(11 a + 31\right)\cdot 43^{2} + \left(5 a + 35\right)\cdot 43^{3} + \left(27 a + 33\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 + 14\cdot 43 + 4\cdot 43^{2} + 16\cdot 43^{3} + 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3)(4,5)$ |
| $(1,2)(3,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,4)(3,5)$ |
$-2$ |
| $3$ |
$2$ |
$(1,2)(3,5)(4,6)$ |
$0$ |
| $3$ |
$2$ |
$(1,5)(3,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,4,5)(2,3,6)$ |
$-1$ |
| $2$ |
$6$ |
$(1,3,4,6,5,2)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
