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:
| $r_{ 1 }$ |
$=$ |
$ 32 a + 1 + \left(37 a + 13\right)\cdot 43 + \left(27 a + 21\right)\cdot 43^{2} + \left(28 a + 13\right)\cdot 43^{3} + \left(40 a + 10\right)\cdot 43^{4} + \left(25 a + 19\right)\cdot 43^{5} + \left(21 a + 30\right)\cdot 43^{6} + \left(35 a + 21\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 29\cdot 43 + 32\cdot 43^{2} + 19\cdot 43^{3} + 4\cdot 43^{4} + 24\cdot 43^{5} + 21\cdot 43^{6} + 24\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 + 30\cdot 43 + 3\cdot 43^{2} + 25\cdot 43^{3} + 13\cdot 43^{4} + 39\cdot 43^{5} + 20\cdot 43^{6} + 31\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 a + 32 + 11 a\cdot 43 + \left(37 a + 17\right)\cdot 43^{2} + \left(20 a + 36\right)\cdot 43^{3} + 11 a\cdot 43^{4} + \left(8 a + 1\right)\cdot 43^{5} + \left(9 a + 36\right)\cdot 43^{6} + \left(39 a + 35\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 a + 33 + \left(5 a + 18\right)\cdot 43 + \left(15 a + 11\right)\cdot 43^{2} + \left(14 a + 14\right)\cdot 43^{3} + \left(2 a + 22\right)\cdot 43^{4} + \left(17 a + 4\right)\cdot 43^{5} + \left(21 a + 26\right)\cdot 43^{6} + \left(7 a + 35\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 24 a + 8 + \left(31 a + 36\right)\cdot 43 + \left(5 a + 42\right)\cdot 43^{2} + \left(22 a + 19\right)\cdot 43^{3} + \left(31 a + 34\right)\cdot 43^{4} + \left(34 a + 40\right)\cdot 43^{5} + \left(33 a + 36\right)\cdot 43^{6} + \left(3 a + 22\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,6)$ |
| $(1,2,5)(3,6,4)$ |
| $(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,6)$ |
$-3$ |
| $3$ |
$2$ |
$(1,4)$ |
$1$ |
| $3$ |
$2$ |
$(1,4)(2,3)$ |
$-1$ |
| $6$ |
$2$ |
$(2,5)(3,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$1$ |
| $8$ |
$3$ |
$(1,2,5)(3,6,4)$ |
$0$ |
| $6$ |
$4$ |
$(1,3,4,2)$ |
$-1$ |
| $6$ |
$4$ |
$(1,6,4,5)(2,3)$ |
$1$ |
| $8$ |
$6$ |
$(1,3,6,4,2,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
