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:
| $r_{ 1 }$ |
$=$ |
$ 37 + 28\cdot 43 + 29\cdot 43^{2} + 14\cdot 43^{3} + 14\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 7 + \left(14 a + 24\right)\cdot 43 + \left(11 a + 21\right)\cdot 43^{2} + \left(17 a + 37\right)\cdot 43^{3} + \left(19 a + 28\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 34 a + 38 + \left(14 a + 12\right)\cdot 43 + \left(a + 36\right)\cdot 43^{2} + \left(11 a + 11\right)\cdot 43^{3} + \left(33 a + 16\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 a + 19 + \left(28 a + 26\right)\cdot 43 + \left(31 a + 18\right)\cdot 43^{2} + 25 a\cdot 43^{3} + \left(23 a + 31\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a + 29 + \left(28 a + 36\right)\cdot 43 + \left(41 a + 22\right)\cdot 43^{2} + \left(31 a + 21\right)\cdot 43^{3} + \left(9 a + 38\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $10$ |
$2$ |
$(1,2)$ |
$2$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
