Galois orbit of Artin representations 4.2e8_17e4.8t26.2

• Label: 4.2e8_17e4.8t26.2
• Dimension: 4
• Group: $(C_4^2 : C_2):C_2$
• Conductor: $ 2^{8} \cdot 17^{4}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(C_4^2 : C_2):C_2$
Conductor:	$21381376= 2^{8} \cdot 17^{4} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + x^{6} + 2 x^{5} - 2 x^{4} - 2 x^{3} + x^{2} + 2 x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(C_4^2 : C_2):C_2$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 457 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 74 + 104\cdot 457 + 69\cdot 457^{2} + 319\cdot 457^{3} + 332\cdot 457^{4} + 204\cdot 457^{5} +O\left(457^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 88 + 47\cdot 457 + 152\cdot 457^{2} + 198\cdot 457^{3} + 236\cdot 457^{4} + 203\cdot 457^{5} +O\left(457^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 103 + 439\cdot 457 + 359\cdot 457^{2} + 290\cdot 457^{3} + 202\cdot 457^{4} + 231\cdot 457^{5} +O\left(457^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 146 + 418\cdot 457 + 14\cdot 457^{2} + 198\cdot 457^{3} + 65\cdot 457^{4} + 182\cdot 457^{5} +O\left(457^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 296 + 345\cdot 457 + 342\cdot 457^{2} + 350\cdot 457^{3} + 193\cdot 457^{4} + 155\cdot 457^{5} +O\left(457^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 352 + 400\cdot 457 + 139\cdot 457^{2} + 302\cdot 457^{3} + 139\cdot 457^{4} + 294\cdot 457^{5} +O\left(457^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 385 + 102\cdot 457 + 404\cdot 457^{2} + 166\cdot 457^{3} + 418\cdot 457^{4} + 372\cdot 457^{5} +O\left(457^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 386 + 426\cdot 457 + 344\cdot 457^{2} + 457^{3} + 239\cdot 457^{4} + 183\cdot 457^{5} +O\left(457^{ 6 }\right)$

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

Cycle notation

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

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

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

$(1,6)(2,5)$

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

$(3,8)(4,7)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.