Galois orbit of Artin representations 4.2e4_3e2_5e2_7e2.8t22.9

• Label: 4.2e4_3e2_5e2_7e2.8t22.9
• 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} - 4 x^{5} - 12 x^{4} + 16 x^{3} + 43 x^{2} + 30 x + 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 $\Q_{ 709 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 8 + 368\cdot 709 + 530\cdot 709^{2} + 248\cdot 709^{3} + 116\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 291 + 462\cdot 709 + 641\cdot 709^{2} + 370\cdot 709^{3} + 549\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 425 + 475\cdot 709 + 518\cdot 709^{2} + 117\cdot 709^{3} + 658\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 469 + 702\cdot 709 + 686\cdot 709^{2} + 553\cdot 709^{3} + 616\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 498 + 237\cdot 709 + 118\cdot 709^{2} + 163\cdot 709^{3} + 451\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 528 + 613\cdot 709 + 685\cdot 709^{2} + 180\cdot 709^{3} + 258\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 639 + 269\cdot 709 + 581\cdot 709^{2} + 646\cdot 709^{3} + 312\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 689 + 414\cdot 709 + 490\cdot 709^{2} + 553\cdot 709^{3} + 581\cdot 709^{4} +O\left(709^{ 5 }\right)$

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

Cycle notation

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

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

$(2,8)(3,4)$

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

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

$(1,7)(2,8)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.