Galois orbit of Artin representations 4.2e6_3e2_5e2_13e2.8t22.13

• Label: 4.2e6_3e2_5e2_13e2.8t22.13
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$2433600= 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2} $
Artin number field:	Splitting field of $f= x^{8} - 7 x^{6} - 10 x^{5} + 3 x^{4} + 20 x^{3} + 63 x^{2} + 110 x + 76 $ 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 $\Q_{ 199 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 46 + 46\cdot 199 + 189\cdot 199^{2} + 111\cdot 199^{3} + 199^{4} +O\left(199^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 70 + 77\cdot 199 + 60\cdot 199^{2} + 150\cdot 199^{3} + 13\cdot 199^{4} +O\left(199^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 84 + 104\cdot 199 + 184\cdot 199^{2} + 128\cdot 199^{3} + 29\cdot 199^{4} +O\left(199^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 102 + 78\cdot 199 + 92\cdot 199^{2} + 80\cdot 199^{3} + 62\cdot 199^{4} +O\left(199^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 113 + 50\cdot 199 + 57\cdot 199^{2} + 38\cdot 199^{3} + 7\cdot 199^{4} +O\left(199^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 117 + 30\cdot 199 + 178\cdot 199^{2} + 178\cdot 199^{3} + 6\cdot 199^{4} +O\left(199^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 122 + 71\cdot 199 + 172\cdot 199^{2} + 68\cdot 199^{3} + 183\cdot 199^{4} +O\left(199^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 142 + 137\cdot 199 + 60\cdot 199^{2} + 38\cdot 199^{3} + 93\cdot 199^{4} +O\left(199^{ 5 }\right)$

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

Cycle notation

$(1,7)(2,3)$

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

$(1,7)(4,8)$

$(1,7)(5,6)$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.