Galois group: 3T1

• Label: 3T1
• Degree: $3$
• Order: $3$
• Cyclic: yes
• Abelian: yes
• Solvable: yes
• Primitive: yes
• $p$-group: yes
• Group: $C_3$

Group invariants

Abstract group:		$C_3$
Order:		$3$ (is prime)
Cyclic:		yes
Abelian:		yes
Solvable:		yes
Nilpotency class:		$1$

Group action invariants

Degree $n$:		$3$
Transitive number $t$:		$1$
CHM label:		$A3$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$3$
Generators:		$(1,2,3)$

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

Label	Cycle Type	Size	Order	Index	Representative
1A	$1^{3}$	$1$	$1$	$0$	$()$
3A1	$3$	$1$	$3$	$2$	$(1,2,3)$
3A-1	$3$	$1$	$3$	$2$	$(1,3,2)$

Malle's constant $a(G)$:     $1/2$

Character table

		1A	3A1	3A-1
	Size	1	1	1
	3 P	1A	3A-1	3A1
	Type
3.1.1a	R	$1$	$1$	$1$
3.1.1b1	C	$1$	${\zeta }_3^{-1}$	${\zeta }_3$
3.1.1b2	C	$1$	${\zeta }_3$	${\zeta }_3^{-1}$

Indecomposable integral representations

Complete list of indecomposable integral representations:

Name	Dim	$(1,2,3) \mapsto $
Triv	$1$	$\left(\begin{array}{r}1\end{array}\right)$
$J$	$2$	$\left(\begin{array}{rr}0 & 1\\-1 & -1\end{array}\right)$
$R$	$3$	$\left(\begin{array}{rrr}0 & 1 & 0\\0 & 0 & 1\\1 & 0 & 0\end{array}\right)$

The decomposition of an arbitrary integral representation as a direct sum of indecomposables is unique.

Regular extensions

$f_{ 1 } =$	$x^{3}-t x^{2}+\left(t-3\right) x+1$
	The polynomial $f_{1}$ is generic for any base field $K$