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:
| $r_{ 1 }$ |
$=$ |
$ 15 a + 16 + \left(36 a + 2\right)\cdot 43 + \left(15 a + 28\right)\cdot 43^{2} + \left(37 a + 6\right)\cdot 43^{3} + \left(16 a + 27\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 a + 31 + \left(6 a + 23\right)\cdot 43 + \left(27 a + 7\right)\cdot 43^{2} + \left(5 a + 28\right)\cdot 43^{3} + \left(26 a + 6\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 + 41\cdot 43 + 28\cdot 43^{2} + 16\cdot 43^{3} + 34\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 + 33\cdot 43 + 13\cdot 43^{2} + 25\cdot 43^{3} + 31\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 a + 27 + \left(21 a + 39\right)\cdot 43 + \left(30 a + 20\right)\cdot 43^{2} + \left(28 a + 5\right)\cdot 43^{3} + \left(6 a + 4\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 13 a + 14 + \left(21 a + 31\right)\cdot 43 + \left(12 a + 29\right)\cdot 43^{2} + \left(14 a + 3\right)\cdot 43^{3} + \left(36 a + 25\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(4,5,6)$ |
| $(1,4)(2,5)(3,6)$ |
| $(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$0$ |
| $6$ |
$2$ |
$(2,3)$ |
$-2$ |
| $9$ |
$2$ |
$(2,3)(5,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-2$ |
| $4$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $18$ |
$4$ |
$(1,4)(2,6,3,5)$ |
$0$ |
| $12$ |
$6$ |
$(1,5,2,6,3,4)$ |
$0$ |
| $12$ |
$6$ |
$(2,3)(4,5,6)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
