Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 a + 12 + 13\cdot 43 + \left(11 a + 12\right)\cdot 43^{2} + \left(23 a + 32\right)\cdot 43^{3} + 35 a\cdot 43^{4} + \left(15 a + 19\right)\cdot 43^{5} + \left(11 a + 37\right)\cdot 43^{6} + \left(7 a + 25\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 32\cdot 43 + 7\cdot 43^{2} + 9\cdot 43^{3} + 29\cdot 43^{4} + 24\cdot 43^{5} + 15\cdot 43^{6} + 38\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 a + 20 + \left(21 a + 23\right)\cdot 43 + \left(17 a + 22\right)\cdot 43^{2} + \left(23 a + 22\right)\cdot 43^{3} + \left(31 a + 2\right)\cdot 43^{4} + \left(10 a + 31\right)\cdot 43^{5} + \left(38 a + 42\right)\cdot 43^{6} + \left(28 a + 40\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 14 + 7\cdot 43 + 2\cdot 43^{2} + 35\cdot 43^{3} + 29\cdot 43^{4} + 43^{5} + 16\cdot 43^{6} + 13\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a + 10 + \left(21 a + 12\right)\cdot 43 + \left(25 a + 18\right)\cdot 43^{2} + \left(19 a + 28\right)\cdot 43^{3} + \left(11 a + 10\right)\cdot 43^{4} + \left(32 a + 10\right)\cdot 43^{5} + \left(4 a + 27\right)\cdot 43^{6} + \left(14 a + 31\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 27 a + 28 + \left(42 a + 40\right)\cdot 43 + \left(31 a + 22\right)\cdot 43^{2} + \left(19 a + 1\right)\cdot 43^{3} + \left(7 a + 13\right)\cdot 43^{4} + \left(27 a + 42\right)\cdot 43^{5} + \left(31 a + 32\right)\cdot 43^{6} + \left(35 a + 21\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(1,3,2,5,6,4)$ |
| $(2,6)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,4)(3,6)$ |
$-2$ |
| $3$ |
$2$ |
$(1,2)(4,5)$ |
$0$ |
| $3$ |
$2$ |
$(1,5)(2,3)(4,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,2,6)(3,5,4)$ |
$-1$ |
| $2$ |
$6$ |
$(1,3,2,5,6,4)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
