Galois orbit of Artin representations 4.2e4_3e2_5e2_7e2.8t22.2

• Label: 4.2e4_3e2_5e2_7e2.8t22.2
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$176400= 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{8} - x^{7} + x^{6} + 5 x^{5} - 5 x^{3} + x^{2} + x + 1 $ 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 $\Q_{ 709 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 5 + 477\cdot 709 + 563\cdot 709^{2} + 290\cdot 709^{3} + 469\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 46 + 123\cdot 709 + 271\cdot 709^{2} + 294\cdot 709^{3} + 222\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 84 + 615\cdot 709 + 80\cdot 709^{2} + 644\cdot 709^{3} + 287\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 211 + 565\cdot 709 + 192\cdot 709^{2} + 492\cdot 709^{3} + 188\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 262 + 640\cdot 709 + 427\cdot 709^{2} + 159\cdot 709^{3} + 357\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 314 + 567\cdot 709 + 118\cdot 709^{2} + 615\cdot 709^{3} + 365\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 567 + 75\cdot 709 + 108\cdot 709^{2} + 306\cdot 709^{3} + 157\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 639 + 480\cdot 709 + 363\cdot 709^{2} + 33\cdot 709^{3} + 78\cdot 709^{4} +O\left(709^{ 5 }\right)$

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

Cycle notation

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

$(2,5)(3,4)$

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

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

$(1,7)(2,5)$

$(2,5)(6,8)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.