Galois orbit of Artin representations 4.2e10_3e2_7e2.8t22.6

• Label: 4.2e10_3e2_7e2.8t22.6
• 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} - 2 x^{6} + 2 x^{4} + 6 x^{2} + 9 $ 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:

$r_{ 1 }$	$=$	$ 2 + 136\cdot 193 + 49\cdot 193^{2} + 74\cdot 193^{3} + 192\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 12 + 155\cdot 193 + 70\cdot 193^{2} + 65\cdot 193^{3} + 52\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 18 + 26\cdot 193 + 159\cdot 193^{2} + 137\cdot 193^{3} + 131\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 63 + 101\cdot 193 + 111\cdot 193^{2} + 2\cdot 193^{3} + 87\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 130 + 91\cdot 193 + 81\cdot 193^{2} + 190\cdot 193^{3} + 105\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 175 + 166\cdot 193 + 33\cdot 193^{2} + 55\cdot 193^{3} + 61\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 181 + 37\cdot 193 + 122\cdot 193^{2} + 127\cdot 193^{3} + 140\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 191 + 56\cdot 193 + 143\cdot 193^{2} + 118\cdot 193^{3} +O\left(193^{ 5 }\right)$

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

Cycle notation

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

$(3,6)(4,5)$

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

$(1,8)(4,5)$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.