Galois group: 10T4

• Label: 10T4
• Degree: $10$
• Order: $20$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $F_5$

Group invariants

Abstract group:		$F_5$
Order:		$20=2^{2} \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: $F_5$

Low degree siblings

5T3, 20T5

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

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	4A1	4A-1	5A
	Size	1	5	5	5	4
	2 P	1A	1A	2A	2A	5A
	5 P	1A	2A	4A1	4A-1	1A
	Type
20.3.1a	R	$1$	$1$	$1$	$1$	$1$
20.3.1b	R	$1$	$1$	$-1$	$-1$	$1$
20.3.1c1	C	$1$	$-1$	$-i$	$i$	$1$
20.3.1c2	C	$1$	$-1$	$i$	$-i$	$1$
20.3.4a	R	$4$	$0$	$0$	$0$	$-1$

Regular extensions

$f_{ 1 } =$

$x^{10} - 5 x^{9} + \left(-t^{2} + t + 17\right) x^{8} + \left(4 t^{2} - 4 t - 38\right) x^{7} + \left(-8 t^{2} + 3 t + 66\right) x^{6} + \left(10 t^{2} + 5 t - 86\right) x^{5} + \left(-9 t^{2} - 9 t + 69\right) x^{4} + \left(6 t^{2} + 5 t - 29\right) x^{3} + \left(-2 t^{2} - 3 t + 4\right) x^{2} + \left(2 t + 1\right) x - 1$