Galois orbit of Artin representations 4.2e4_3e4_5e2_7e2.8t22.8

• Label: 4.2e4_3e4_5e2_7e2.8t22.8
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$1587600= 2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{8} - x^{7} - 4 x^{6} - x^{5} - 2 x^{4} + x^{3} + 29 x^{2} + 40 x + 16 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3 : D_4 $
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 12 + 14\cdot 109 + 83\cdot 109^{2} + 42\cdot 109^{3} + 4\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 17 + 53\cdot 109 + 76\cdot 109^{2} + 7\cdot 109^{3} + 44\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 18 + 59\cdot 109 + 54\cdot 109^{2} + 86\cdot 109^{3} + 2\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 47 + 45\cdot 109 + 37\cdot 109^{2} + 7\cdot 109^{3} + 90\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 50 + 55\cdot 109 + 68\cdot 109^{2} + 79\cdot 109^{3} + 24\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 54 + 24\cdot 109 + 101\cdot 109^{2} + 35\cdot 109^{3} + 59\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 59 + 68\cdot 109 + 52\cdot 109^{2} + 78\cdot 109^{3} + 89\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 71 + 6\cdot 109 + 71\cdot 109^{2} + 97\cdot 109^{3} + 11\cdot 109^{4} +O\left(109^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation

$(2,5)(3,6)$

$(1,4,8,7)(2,6,5,3)$

$(1,8)(2,5)$

$(1,5)(2,8)(3,4)(6,7)$

$(1,4,8,7)(2,3,5,6)$

$(2,5)(4,7)$

$(1,2)(3,4)(5,8)(6,7)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$
$1$	$1$	$()$	$4$
$1$	$2$	$(1,8)(2,5)(3,6)(4,7)$	$-4$
$2$	$2$	$(1,8)(2,5)$	$0$
$2$	$2$	$(1,5)(2,8)(3,4)(6,7)$	$0$
$2$	$2$	$(1,2)(3,4)(5,8)(6,7)$	$0$
$2$	$2$	$(1,6)(2,7)(3,8)(4,5)$	$0$
$2$	$2$	$(2,5)(3,6)$	$0$
$2$	$2$	$(1,8)(3,6)$	$0$
$2$	$2$	$(1,4)(2,3)(5,6)(7,8)$	$0$
$2$	$2$	$(1,6)(2,4)(3,8)(5,7)$	$0$
$2$	$2$	$(1,4)(2,6)(3,5)(7,8)$	$0$
$2$	$4$	$(1,4,8,7)(2,6,5,3)$	$0$
$2$	$4$	$(1,3,8,6)(2,7,5,4)$	$0$
$2$	$4$	$(1,4,8,7)(2,3,5,6)$	$0$
$2$	$4$	$(1,5,8,2)(3,7,6,4)$	$0$
$2$	$4$	$(1,2,8,5)(3,7,6,4)$	$0$
$2$	$4$	$(1,3,8,6)(2,4,5,7)$	$0$

The blue line marks the conjugacy class containing complex conjugation.