Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 13\cdot 43 + 34\cdot 43^{2} + 31\cdot 43^{3} + 12\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 36 a + 23 + \left(30 a + 9\right)\cdot 43 + \left(26 a + 38\right)\cdot 43^{2} + \left(14 a + 9\right)\cdot 43^{3} + \left(14 a + 4\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a + 20 + \left(3 a + 30\right)\cdot 43 + \left(15 a + 26\right)\cdot 43^{2} + \left(5 a + 6\right)\cdot 43^{3} + \left(17 a + 5\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 16 + \left(12 a + 4\right)\cdot 43 + \left(16 a + 34\right)\cdot 43^{2} + \left(28 a + 40\right)\cdot 43^{3} + \left(28 a + 3\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 a + 25 + \left(39 a + 28\right)\cdot 43 + \left(27 a + 38\right)\cdot 43^{2} + \left(37 a + 39\right)\cdot 43^{3} + \left(25 a + 16\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $12$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $12$ |
$5$ |
$(1,3,4,5,2)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
