Galois orbit of Artin representations 4.2e4_3e2_5e2_7e2.8t22.4

• Label: 4.2e4_3e2_5e2_7e2.8t22.4
• 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} - 2 x^{7} - 3 x^{6} + 10 x^{5} - 10 x^{4} - 4 x^{3} + 10 x^{2} + 6 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:

$r_{ 1 }$	$=$	$ 14 + 387\cdot 709 + 444\cdot 709^{2} + 677\cdot 709^{3} + 365\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 51 + 600\cdot 709 + 125\cdot 709^{2} + 585\cdot 709^{3} + 691\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 120 + 7\cdot 709 + 536\cdot 709^{2} + 465\cdot 709^{3} + 514\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 141 + 337\cdot 709 + 556\cdot 709^{2} + 525\cdot 709^{3} + 266\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 267 + 408\cdot 709 + 282\cdot 709^{2} + 450\cdot 709^{3} + 117\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 398 + 473\cdot 709 + 199\cdot 709^{2} + 550\cdot 709^{3} + 653\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 525 + 423\cdot 709 + 311\cdot 709^{2} + 398\cdot 709^{3} + 554\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 613 + 198\cdot 709 + 379\cdot 709^{2} + 600\cdot 709^{3} + 379\cdot 709^{4} +O\left(709^{ 5 }\right)$

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

Cycle notation

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

$(1,7)(4,6)$

$(1,7)(5,8)$

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.