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:
| $r_{ 1 }$ |
$=$ |
$ 29 a + 36 + \left(36 a + 31\right)\cdot 43 + \left(34 a + 8\right)\cdot 43^{2} + \left(10 a + 23\right)\cdot 43^{3} + \left(22 a + 23\right)\cdot 43^{4} + \left(2 a + 14\right)\cdot 43^{5} + \left(41 a + 24\right)\cdot 43^{6} + \left(32 a + 36\right)\cdot 43^{7} + \left(31 a + 20\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 28 + 2\cdot 43 + 9\cdot 43^{2} + 24\cdot 43^{3} + 33\cdot 43^{4} + 29\cdot 43^{5} + 35\cdot 43^{7} + 24\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 a + 22 + \left(6 a + 39\right)\cdot 43 + \left(8 a + 6\right)\cdot 43^{2} + \left(32 a + 42\right)\cdot 43^{3} + \left(20 a + 34\right)\cdot 43^{4} + \left(40 a + 37\right)\cdot 43^{5} + \left(a + 19\right)\cdot 43^{6} + \left(10 a + 28\right)\cdot 43^{7} + \left(11 a + 19\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + 5 + 23\cdot 43 + \left(29 a + 2\right)\cdot 43^{2} + \left(25 a + 11\right)\cdot 43^{3} + \left(11 a + 33\right)\cdot 43^{4} + \left(6 a + 30\right)\cdot 43^{5} + \left(14 a + 38\right)\cdot 43^{6} + \left(21 a + 21\right)\cdot 43^{7} + \left(10 a + 14\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 37 a + 11 + \left(42 a + 17\right)\cdot 43 + \left(13 a + 31\right)\cdot 43^{2} + \left(17 a + 7\right)\cdot 43^{3} + \left(31 a + 19\right)\cdot 43^{4} + \left(36 a + 25\right)\cdot 43^{5} + \left(28 a + 3\right)\cdot 43^{6} + \left(21 a + 29\right)\cdot 43^{7} + \left(32 a + 3\right)\cdot 43^{8} +O\left(43^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 + 14\cdot 43 + 27\cdot 43^{2} + 20\cdot 43^{3} + 27\cdot 43^{4} + 33\cdot 43^{5} + 41\cdot 43^{6} + 20\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 |
| $(2,5)(3,6)$ |
| $(1,2)(3,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,6)(3,5)$ |
$-2$ |
| $3$ |
$2$ |
$(1,2)(3,5)(4,6)$ |
$0$ |
| $3$ |
$2$ |
$(1,3)(4,5)$ |
$0$ |
| $2$ |
$3$ |
$(1,6,3)(2,5,4)$ |
$-1$ |
| $2$ |
$6$ |
$(1,5,6,4,3,2)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
