Galois group: 5T1

• Label: 5T1
• Degree: $5$
• Order: $5$
• Cyclic: yes
• Abelian: yes
• Solvable: yes
• Primitive: yes
• $p$-group: yes
• Group: $C_5$

Group action invariants

Degree $n$:		$5$
Transitive number $t$:		$1$
Group:		$C_5$
CHM label:		$C(5) = 5$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$5$
Generators:		(1,2,3,4,5)

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	$1^{5}$	$1$	$1$	$0$	$()$
5A1	$5$	$1$	$5$	$4$	$(1,2,3,4,5)$
5A-1	$5$	$1$	$5$	$4$	$(1,4,2,5,3)$
5A2	$5$	$1$	$5$	$4$	$(1,5,4,3,2)$
5A-2	$5$	$1$	$5$	$4$	$(1,3,5,2,4)$

Malle's constant $a(G)$:     $1/4$

Group invariants

Order:		$5$ (is prime)
Cyclic:		yes
Abelian:		yes
Solvable:		yes
Nilpotency class:		$1$
Label:		5.1
Character table:

		1A	5A1	5A-1	5A2	5A-2
	Size	1	1	1	1	1
	5 P	1A	5A2	5A1	5A-2	5A-1
	Type
5.1.1a	R	$1$	$1$	$1$	$1$	$1$
5.1.1b1	C	$1$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-1}$
5.1.1b2	C	$1$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5$
5.1.1b3	C	$1$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^2$
5.1.1b4	C	$1$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5^{-2}$

Indecomposable integral representations

Complete list of indecomposable integral representations:

Name	Dim	$(1,2,3,4,5) \mapsto $
Triv	$1$	$\left(\begin{array}{r}1\end{array}\right)$
$J$	$4$	$\left(\begin{array}{rrrr}0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\\-1 & -1 & -1 & -1\end{array}\right)$
$R$	$5$	$\left(\begin{array}{rrrrr}0 & 1 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & 1\\1 & 0 & 0 & 0 & 0\end{array}\right)$

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