Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 8\cdot 43 + 25\cdot 43^{2} + 9\cdot 43^{3} + 11\cdot 43^{4} + 29\cdot 43^{5} + 27\cdot 43^{6} + 33\cdot 43^{7} + 2\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 41 a + 37 + \left(15 a + 1\right)\cdot 43 + \left(15 a + 30\right)\cdot 43^{2} + \left(17 a + 25\right)\cdot 43^{3} + \left(5 a + 24\right)\cdot 43^{4} + \left(20 a + 25\right)\cdot 43^{5} + \left(39 a + 19\right)\cdot 43^{6} + \left(25 a + 34\right)\cdot 43^{7} + \left(34 a + 20\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a + 30 + \left(16 a + 20\right)\cdot 43 + \left(25 a + 42\right)\cdot 43^{2} + \left(23 a + 39\right)\cdot 43^{3} + \left(11 a + 24\right)\cdot 43^{4} + 38\cdot 43^{5} + \left(29 a + 18\right)\cdot 43^{6} + \left(21 a + 37\right)\cdot 43^{7} + \left(42 a + 4\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a + 35 + \left(27 a + 19\right)\cdot 43 + \left(27 a + 29\right)\cdot 43^{2} + \left(25 a + 27\right)\cdot 43^{3} + \left(37 a + 12\right)\cdot 43^{4} + \left(22 a + 40\right)\cdot 43^{5} + \left(3 a + 38\right)\cdot 43^{6} + \left(17 a + 20\right)\cdot 43^{7} + \left(8 a + 29\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 a + 40 + \left(26 a + 26\right)\cdot 43 + \left(17 a + 8\right)\cdot 43^{2} + \left(19 a + 38\right)\cdot 43^{3} + \left(31 a + 12\right)\cdot 43^{4} + \left(42 a + 27\right)\cdot 43^{5} + \left(13 a + 4\right)\cdot 43^{6} + \left(21 a + 30\right)\cdot 43^{7} + 25\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 + 8\cdot 43 + 36\cdot 43^{2} + 30\cdot 43^{3} + 42\cdot 43^{4} + 10\cdot 43^{5} + 19\cdot 43^{6} + 15\cdot 43^{7} + 2\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)$ |
| $(1,2)(5,6)$ |
| $(1,3,2)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,4)$ |
$-3$ |
| $3$ |
$2$ |
$(1,6)(2,5)$ |
$-1$ |
| $3$ |
$2$ |
$(2,5)$ |
$1$ |
| $6$ |
$2$ |
$(1,2)(5,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,6)(2,3)(4,5)$ |
$1$ |
| $8$ |
$3$ |
$(1,3,2)(4,5,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,5,6,2)$ |
$-1$ |
| $6$ |
$4$ |
$(1,6)(2,4,5,3)$ |
$1$ |
| $8$ |
$6$ |
$(1,3,2,6,4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
