Galois orbit of Artin representations 4.2e10_3e2_7e2.8t22.5

• Label: 4.2e10_3e2_7e2.8t22.5
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{10} \cdot 3^{2} \cdot 7^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$451584= 2^{10} \cdot 3^{2} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{8} + 8 x^{6} - 4 x^{5} + 15 x^{4} - 16 x^{3} + 12 x^{2} - 16 x + 7 $ 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 $\Q_{ 193 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 12 + 161\cdot 193 + 167\cdot 193^{2} + 29\cdot 193^{3} + 60\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 80 + 188\cdot 193 + 152\cdot 193^{2} + 126\cdot 193^{3} + 133\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 101 + 172\cdot 193 + 62\cdot 193^{2} + 136\cdot 193^{3} + 159\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 110 + 61\cdot 193 + 95\cdot 193^{2} + 182\cdot 193^{3} + 89\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 132 + 156\cdot 193 + 76\cdot 193^{2} + 163\cdot 193^{3} + 146\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 163 + 183\cdot 193 + 59\cdot 193^{2} + 37\cdot 193^{3} + 76\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 176 + 25\cdot 193 + 132\cdot 193^{2} + 80\cdot 193^{3} + 113\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 191 + 14\cdot 193 + 24\cdot 193^{2} + 15\cdot 193^{3} + 185\cdot 193^{4} +O\left(193^{ 5 }\right)$

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

Cycle notation

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

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

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

$(1,4)(2,5)$

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

$(2,5)(7,8)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.