Galois group: 10T1

• Label: 10T1
• Degree: $10$
• Order: $10$
• Cyclic: yes
• Abelian: yes
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_{10}$

Group action invariants

Degree $n$:		$10$
Transitive number $t$:		$1$
Group:		$C_{10}$
CHM label:		$C(10)=5[x]2$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$10$
Generators:		(1,2,3,4,5,6,7,8,9,10)

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$5$:	$C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: $C_5$

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	$1^{10}$	$1$	$1$	$0$	$()$
2A	$2^{5}$	$1$	$2$	$5$	$( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$
5A1	$5^{2}$	$1$	$5$	$8$	$( 1, 9, 7, 5, 3)( 2,10, 8, 6, 4)$
5A-1	$5^{2}$	$1$	$5$	$8$	$( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
5A2	$5^{2}$	$1$	$5$	$8$	$( 1, 7, 3, 9, 5)( 2, 8, 4,10, 6)$
5A-2	$5^{2}$	$1$	$5$	$8$	$( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)$
10A1	$10$	$1$	$10$	$9$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$
10A-1	$10$	$1$	$10$	$9$	$( 1,10, 9, 8, 7, 6, 5, 4, 3, 2)$
10A3	$10$	$1$	$10$	$9$	$( 1, 4, 7,10, 3, 6, 9, 2, 5, 8)$
10A-3	$10$	$1$	$10$	$9$	$( 1, 8, 5, 2, 9, 6, 3,10, 7, 4)$

Malle's constant $a(G)$:     $1/5$

Group invariants

Order:		$10=2 \cdot 5$
Cyclic:		yes
Abelian:		yes
Solvable:		yes
Nilpotency class:		$1$
Label:		10.2
Character table:

		1A	2A	5A1	5A-1	5A2	5A-2	10A1	10A-1	10A3	10A-3
	Size	1	1	1	1	1	1	1	1	1	1
	2 P	1A	1A	5A-2	5A2	5A1	5A-1	5A1	5A-1	5A-2	5A2
	5 P	1A	2A	1A	1A	1A	1A	2A	2A	2A	2A
	Type
10.2.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
10.2.1b	R	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
10.2.1c1	C	$1$	$1$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^2$	${\zeta }_5^{-2}$
10.2.1c2	C	$1$	$1$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^{-2}$	${\zeta }_5^2$
10.2.1c3	C	$1$	$1$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5$	${\zeta }_5^{-1}$
10.2.1c4	C	$1$	$1$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5^{-1}$	${\zeta }_5$
10.2.1d1	C	$1$	$-1$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-1}$	$-{\zeta }_5^{-1}$	$-{\zeta }_5$	$-{\zeta }_5^2$	$-{\zeta }_5^{-2}$
10.2.1d2	C	$1$	$-1$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5$	$-{\zeta }_5$	$-{\zeta }_5^{-1}$	$-{\zeta }_5^{-2}$	$-{\zeta }_5^2$
10.2.1d3	C	$1$	$-1$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^2$	$-{\zeta }_5^2$	$-{\zeta }_5^{-2}$	$-{\zeta }_5$	$-{\zeta }_5^{-1}$
10.2.1d4	C	$1$	$-1$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5^{-2}$	$-{\zeta }_5^{-2}$	$-{\zeta }_5^2$	$-{\zeta }_5^{-1}$	$-{\zeta }_5$