Galois group: 13T2

• Label: 13T2
• Degree: $13$
• Order: $26$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $D_{13}$

Group invariants

Abstract group:		$D_{13}$
Order:		$26=2 \cdot 13$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$13$
Transitive number $t$:		$2$
CHM label:		$D(13)=13:2$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,2,3,4,5,6,7,8,9,10,11,12,13)$, $(1,12)(2,11)(3,10)(4,9)(5,8)(6,7)$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

26T2

Siblings are shown 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	$1^{13}$	$1$	$1$	$0$	$()$
2A	$2^{6},1$	$13$	$2$	$6$	$( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
13A1	$13$	$2$	$13$	$12$	$( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)$
13A2	$13$	$2$	$13$	$12$	$( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)$
13A3	$13$	$2$	$13$	$12$	$( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)$
13A4	$13$	$2$	$13$	$12$	$( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)$
13A5	$13$	$2$	$13$	$12$	$( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
13A6	$13$	$2$	$13$	$12$	$( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)$

Malle's constant $a(G)$:     $1/6$

Character table

		1A	2A	13A1	13A2	13A3	13A4	13A5	13A6
	Size	1	13	2	2	2	2	2	2
	2 P	1A	1A	13A6	13A5	13A3	13A1	13A2	13A4
	13 P	1A	2A	13A4	13A1	13A2	13A5	13A3	13A6
	Type
26.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
26.1.1b	R	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$
26.1.2a1	R	$2$	$0$	${\zeta }_{13}^{-6}+{\zeta }_{13}^6$	${\zeta }_{13}^{-1}+{\zeta }_{13}$	${\zeta }_{13}^{-5}+{\zeta }_{13}^5$	${\zeta }_{13}^{-2}+{\zeta }_{13}^2$	${\zeta }_{13}^{-4}+{\zeta }_{13}^4$	${\zeta }_{13}^{-3}+{\zeta }_{13}^3$
26.1.2a2	R	$2$	$0$	${\zeta }_{13}^{-5}+{\zeta }_{13}^5$	${\zeta }_{13}^{-3}+{\zeta }_{13}^3$	${\zeta }_{13}^{-2}+{\zeta }_{13}^2$	${\zeta }_{13}^{-6}+{\zeta }_{13}^6$	${\zeta }_{13}^{-1}+{\zeta }_{13}$	${\zeta }_{13}^{-4}+{\zeta }_{13}^4$
26.1.2a3	R	$2$	$0$	${\zeta }_{13}^{-4}+{\zeta }_{13}^4$	${\zeta }_{13}^{-5}+{\zeta }_{13}^5$	${\zeta }_{13}^{-1}+{\zeta }_{13}$	${\zeta }_{13}^{-3}+{\zeta }_{13}^3$	${\zeta }_{13}^{-6}+{\zeta }_{13}^6$	${\zeta }_{13}^{-2}+{\zeta }_{13}^2$
26.1.2a4	R	$2$	$0$	${\zeta }_{13}^{-3}+{\zeta }_{13}^3$	${\zeta }_{13}^{-6}+{\zeta }_{13}^6$	${\zeta }_{13}^{-4}+{\zeta }_{13}^4$	${\zeta }_{13}^{-1}+{\zeta }_{13}$	${\zeta }_{13}^{-2}+{\zeta }_{13}^2$	${\zeta }_{13}^{-5}+{\zeta }_{13}^5$
26.1.2a5	R	$2$	$0$	${\zeta }_{13}^{-2}+{\zeta }_{13}^2$	${\zeta }_{13}^{-4}+{\zeta }_{13}^4$	${\zeta }_{13}^{-6}+{\zeta }_{13}^6$	${\zeta }_{13}^{-5}+{\zeta }_{13}^5$	${\zeta }_{13}^{-3}+{\zeta }_{13}^3$	${\zeta }_{13}^{-1}+{\zeta }_{13}$
26.1.2a6	R	$2$	$0$	${\zeta }_{13}^{-1}+{\zeta }_{13}$	${\zeta }_{13}^{-2}+{\zeta }_{13}^2$	${\zeta }_{13}^{-3}+{\zeta }_{13}^3$	${\zeta }_{13}^{-4}+{\zeta }_{13}^4$	${\zeta }_{13}^{-5}+{\zeta }_{13}^5$	${\zeta }_{13}^{-6}+{\zeta }_{13}^6$

Regular extensions

Data not computed