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:
| $r_{ 1 }$ |
$=$ |
$ 3 + 4\cdot 43 + 19\cdot 43^{2} + 15\cdot 43^{3} + 3\cdot 43^{4} + 6\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 a + 38 + \left(2 a + 35\right)\cdot 43 + \left(16 a + 15\right)\cdot 43^{2} + \left(26 a + 4\right)\cdot 43^{3} + \left(4 a + 18\right)\cdot 43^{4} + \left(25 a + 4\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 a + 39 + \left(19 a + 31\right)\cdot 43 + \left(37 a + 24\right)\cdot 43^{2} + \left(28 a + 20\right)\cdot 43^{3} + \left(27 a + 12\right)\cdot 43^{4} + \left(14 a + 5\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 a + 22 + \left(23 a + 25\right)\cdot 43 + \left(5 a + 42\right)\cdot 43^{2} + \left(14 a + 11\right)\cdot 43^{3} + \left(15 a + 11\right)\cdot 43^{4} + \left(28 a + 35\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 + 10\cdot 43 + 40\cdot 43^{2} + 18\cdot 43^{3} + 43^{4} + 10\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 a + 12 + \left(40 a + 21\right)\cdot 43 + \left(26 a + 29\right)\cdot 43^{2} + \left(16 a + 14\right)\cdot 43^{3} + \left(38 a + 39\right)\cdot 43^{4} + \left(17 a + 24\right)\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,6)(4,5)$ |
| $(3,5,4)$ |
| $(1,3)(2,5)(4,6)$ |
| $(1,2,6)(3,5,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $3$ |
$2$ |
$(1,3)(2,5)(4,6)$ |
$0$ |
| $3$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$0$ |
| $9$ |
$2$ |
$(1,2)(4,5)$ |
$0$ |
| $2$ |
$3$ |
$(1,2,6)(3,5,4)$ |
$-2$ |
| $2$ |
$3$ |
$(1,2,6)(3,4,5)$ |
$-2$ |
| $4$ |
$3$ |
$(1,6,2)$ |
$1$ |
| $6$ |
$6$ |
$(1,5,6,3,2,4)$ |
$0$ |
| $6$ |
$6$ |
$(1,5,6,4,2,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
