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:
| $r_{ 1 }$ |
$=$ |
$ 3 + 43 + 7\cdot 43^{2} + 28\cdot 43^{3} + 25\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 37 a + 2 + \left(5 a + 15\right)\cdot 43 + \left(3 a + 19\right)\cdot 43^{2} + \left(32 a + 14\right)\cdot 43^{3} + \left(29 a + 31\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 a + 1 + \left(32 a + 26\right)\cdot 43 + \left(32 a + 26\right)\cdot 43^{2} + \left(11 a + 3\right)\cdot 43^{3} + \left(21 a + 13\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 + 20\cdot 43 + 32\cdot 43^{2} + 13\cdot 43^{3} + 7\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 a + 39 + \left(37 a + 26\right)\cdot 43 + \left(39 a + 16\right)\cdot 43^{2} + 10 a\cdot 43^{3} + \left(13 a + 29\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 24 a + 20 + \left(10 a + 39\right)\cdot 43 + \left(10 a + 26\right)\cdot 43^{2} + \left(31 a + 25\right)\cdot 43^{3} + \left(21 a + 22\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(3,4)$ |
| $(3,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$0$ |
| $6$ |
$2$ |
$(2,5)$ |
$2$ |
| $9$ |
$2$ |
$(2,5)(4,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,5)(3,4,6)$ |
$-2$ |
| $4$ |
$3$ |
$(1,2,5)$ |
$1$ |
| $18$ |
$4$ |
$(1,3)(2,6,5,4)$ |
$0$ |
| $12$ |
$6$ |
$(1,4,2,6,5,3)$ |
$0$ |
| $12$ |
$6$ |
$(2,5)(3,4,6)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
