Galois orbit of Artin representations 4.2e4_3e2_5e2_31e2.8t22.5

• Label: 4.2e4_3e2_5e2_31e2.8t22.5
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$3459600= 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + 7 x^{6} + 2 x^{5} + 9 x^{4} + 52 x^{3} + 28 x^{2} + 80 x + 100 $ 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 6.

Roots:

$r_{ 1 }$	$=$	$ 15 + 91\cdot 109 + 39\cdot 109^{2} + 55\cdot 109^{3} + 30\cdot 109^{4} + 19\cdot 109^{5} +O\left(109^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 16 + 75\cdot 109 + 100\cdot 109^{2} + 39\cdot 109^{3} + 18\cdot 109^{4} + 31\cdot 109^{5} +O\left(109^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 19 + 107\cdot 109 + 3\cdot 109^{2} + 44\cdot 109^{3} + 101\cdot 109^{4} + 99\cdot 109^{5} +O\left(109^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 20 + 24\cdot 109 + 10\cdot 109^{2} + 31\cdot 109^{3} + 36\cdot 109^{4} + 16\cdot 109^{5} +O\left(109^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 21 + 5\cdot 109 + 72\cdot 109^{2} + 102\cdot 109^{3} + 4\cdot 109^{4} + 72\cdot 109^{5} +O\left(109^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 60 + 53\cdot 109 + 73\cdot 109^{2} + 78\cdot 109^{3} + 67\cdot 109^{4} + 67\cdot 109^{5} +O\left(109^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 76 + 55\cdot 109 + 39\cdot 109^{2} + 39\cdot 109^{3} + 7\cdot 109^{4} + 65\cdot 109^{5} +O\left(109^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 102 + 23\cdot 109 + 96\cdot 109^{2} + 44\cdot 109^{3} + 60\cdot 109^{4} + 64\cdot 109^{5} +O\left(109^{ 6 }\right)$

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

Cycle notation

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

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

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

$(3,6)(4,8)$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.