Galois orbit of Artin representations 4.2e4_3e2_5e2_7e2.8t22.7

• Label: 4.2e4_3e2_5e2_7e2.8t22.7
• 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} - 2 x^{6} + 8 x^{5} - 6 x^{4} - 5 x^{3} + 16 x^{2} - 14 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 $\Q_{ 709 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 4 + 581\cdot 709 + 261\cdot 709^{2} + 403\cdot 709^{3} + 300\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 20 + 485\cdot 709 + 572\cdot 709^{2} + 447\cdot 709^{3} + 477\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 129 + 360\cdot 709 + 656\cdot 709^{2} + 122\cdot 709^{3} + 277\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 133 + 585\cdot 709 + 578\cdot 709^{2} + 702\cdot 709^{3} + 84\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 270 + 553\cdot 709 + 354\cdot 709^{2} + 9\cdot 709^{3} + 504\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 397 + 202\cdot 709 + 279\cdot 709^{2} + 665\cdot 709^{3} + 520\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 514 + 467\cdot 709 + 371\cdot 709^{2} + 701\cdot 709^{3} + 490\cdot 709^{4} +O\left(709^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 661 + 309\cdot 709 + 469\cdot 709^{2} + 491\cdot 709^{3} + 179\cdot 709^{4} +O\left(709^{ 5 }\right)$

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

Cycle notation

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

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

$(1,8)(2,5)$

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.