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:
| $r_{ 1 }$ |
$=$ |
$ 14 a + 12 + \left(37 a + 25\right)\cdot 43 + \left(a + 37\right)\cdot 43^{2} + \left(19 a + 36\right)\cdot 43^{3} + \left(30 a + 1\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 41 a + 36 + \left(33 a + 39\right)\cdot 43 + \left(6 a + 7\right)\cdot 43^{2} + \left(4 a + 17\right)\cdot 43^{3} + 2 a\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 29 a + 26 + \left(5 a + 5\right)\cdot 43 + \left(41 a + 2\right)\cdot 43^{2} + \left(23 a + 11\right)\cdot 43^{3} + \left(12 a + 13\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 + 31\cdot 43 + 4\cdot 43^{2} + 33\cdot 43^{3} + 28\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 2 a + 34 + \left(9 a + 32\right)\cdot 43 + \left(36 a + 23\right)\cdot 43^{2} + \left(38 a + 14\right)\cdot 43^{3} + \left(40 a + 41\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 8 + 37\cdot 43 + 9\cdot 43^{2} + 16\cdot 43^{3} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,4)$ |
| $(1,2)(3,5)(4,6)$ |
| $(1,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,2)(3,5)(4,6)$ |
$0$ |
| $6$ |
$2$ |
$(1,3)$ |
$2$ |
| $9$ |
$2$ |
$(1,3)(2,5)$ |
$0$ |
| $4$ |
$3$ |
$(1,3,4)$ |
$1$ |
| $4$ |
$3$ |
$(1,3,4)(2,5,6)$ |
$-2$ |
| $18$ |
$4$ |
$(1,5,3,2)(4,6)$ |
$0$ |
| $12$ |
$6$ |
$(1,5,3,6,4,2)$ |
$0$ |
| $12$ |
$6$ |
$(1,3)(2,5,6)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
