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 + 6 + \left(25 a + 24\right)\cdot 43 + \left(13 a + 24\right)\cdot 43^{2} + \left(27 a + 33\right)\cdot 43^{3} + \left(12 a + 20\right)\cdot 43^{4} + \left(23 a + 35\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 39 a + 10 + \left(17 a + 2\right)\cdot 43 + \left(29 a + 13\right)\cdot 43^{2} + \left(15 a + 4\right)\cdot 43^{3} + \left(30 a + 6\right)\cdot 43^{4} + \left(19 a + 3\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 1 + \left(4 a + 7\right)\cdot 43 + \left(18 a + 35\right)\cdot 43^{2} + \left(33 a + 3\right)\cdot 43^{3} + \left(18 a + 37\right)\cdot 43^{4} + \left(26 a + 14\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 + 4\cdot 43 + 24\cdot 43^{2} + 16\cdot 43^{3} + 12\cdot 43^{4} + 29\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 + 26\cdot 43^{2} + 8\cdot 43^{3} + 30\cdot 43^{4} + 23\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 a + 8 + \left(38 a + 4\right)\cdot 43 + \left(24 a + 6\right)\cdot 43^{2} + \left(9 a + 19\right)\cdot 43^{3} + \left(24 a + 22\right)\cdot 43^{4} + \left(16 a + 22\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(4,5)$ |
| $(3,6)$ |
| $(1,4,3)(2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$-3$ |
| $3$ |
$2$ |
$(1,2)$ |
$1$ |
| $3$ |
$2$ |
$(1,2)(3,6)$ |
$-1$ |
| $4$ |
$3$ |
$(1,4,3)(2,5,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,3,4)(2,6,5)$ |
$0$ |
| $4$ |
$6$ |
$(1,5,6,2,4,3)$ |
$0$ |
| $4$ |
$6$ |
$(1,3,4,2,6,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
