Properties

Label 2T1
Degree $2$
Order $2$
Cyclic yes
Abelian yes
Solvable yes
Primitive yes
$p$-group yes
Group: $C_2$

Related objects

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Group action invariants

Degree $n$:  $2$
Transitive number $t$:  $1$
Group:  $C_2$
CHM label:  $S2$
Parity:  $-1$
Primitive:  yes
Nilpotency class:  $1$
$|\Aut(F/K)|$:  $2$
Generators:  (1,2)

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Cycle TypeSizeOrderRepresentative
$ 1, 1 $ $1$ $1$ $()$
$ 2 $ $1$ $2$ $(1,2)$

Group invariants

Order:  $2$ (is prime)
Cyclic:  yes
Abelian:  yes
Solvable:  yes
GAP id:  [2, 1]
Character table:   
     2  1  1

       1a 2a
    2P 1a 1a

X.1     1 -1
X.2     1  1

Indecomposable integral representations

Complete list of indecomposable integral representations:

Name Dim $(1,2) \mapsto $
Triv $1$ $\left(\begin{array}{r}1\end{array}\right)$
Sign $1$ $\left(\begin{array}{r}-1\end{array}\right)$
$L$ $2$ $\left(\begin{array}{rr}0 & 1\\1 & 0\end{array}\right)$
The decomposition of an arbitrary integral representation as a direct sum of indecomposables is unique.