Resolvents shown for degrees $\leq 47$
Degree 2: $C_2$
There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
| Cycle Type | Size | Order | Representative |
| $ 1, 1, 1, 1 $ |
$1$ |
$1$ |
$()$ |
| $ 4 $ |
$1$ |
$4$ |
$(1,2,3,4)$ |
| $ 2, 2 $ |
$1$ |
$2$ |
$(1,3)(2,4)$ |
| $ 4 $ |
$1$ |
$4$ |
$(1,4,3,2)$ |
| Character table:
| |
2 2 2 2 2
1a 4a 2a 4b
X.1 1 1 1 1
X.2 1 -1 1 -1
X.3 1 A -1 -A
X.4 1 -A -1 A
A = E(4)
= Sqrt(-1) = i
|
|
Complete
list of indecomposable integral representations:
| Name | Dim |
$(1,2,3,4) \mapsto $ |
|---|
| Triv | $1$ |
$\left(\begin{array}{r}1\end{array}\right)$ |
| $B$ | $1$ |
$\left(\begin{array}{r}-1\end{array}\right)$ |
| $C$ | $2$ |
$\left(\begin{array}{rr}0 & -1\\1 & 0\end{array}\right)$ |
| $E$ | $2$ |
$\left(\begin{array}{rr}0 & 1\\1 & 0\end{array}\right)$ |
| $(C,\text{Triv})$ | $3$ |
$\left(\begin{array}{rrr}1 & 0 & 1\\0 & 0 & -1\\0 & 1 & 0\end{array}\right)$ |
| $(C,B)$ | $3$ |
$\left(\begin{array}{rrr}-1 & 0 & 1\\0 & 0 & -1\\0 & 1 & 0\end{array}\right)$ |
| $(C,E)$ | $4$ |
$\left(\begin{array}{rrrr}0 & 1 & 0 & 0\\1 & 0 & 0 & 1\\0 & 0 & 0 & -1\\0 & 0 & 1 & 0\end{array}\right)$ |
| $(C,E)'$ | $4$ |
$\left(\begin{array}{rrrr}0 & 1 & 0 & 1\\1 & 0 & 0 & -1\\0 & 0 & 0 & -1\\0 & 0 & 1 & 0\end{array}\right)$ |
| $(C,\text{Triv}+B)$ | $4$ |
$\left(\begin{array}{rrrr}1 & 0 & 0 & 1\\0 & -1 & 0 & 1\\0 & 0 & 0 & -1\\0 & 0 & 1 & 0\end{array}\right)$ |
|
The decomposition of an arbitrary integral representation as a direct
sum of indecomposables is unique.
