Galois orbit of Artin representations 4.2e17.8t21.4

• Label: 4.2e17.8t21.4
• Dimension: 4
• Group: $C_2^3: C_4$
• Conductor: $ 2^{17}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3: C_4$
Conductor:	$131072= 2^{17} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{6} + 6 x^{4} - 4 x^{2} + 2 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3: C_4$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 337 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 100 + 162\cdot 337 + 242\cdot 337^{2} + 116\cdot 337^{3} + 212\cdot 337^{4} + 282\cdot 337^{5} +O\left(337^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 109 + 15\cdot 337 + 291\cdot 337^{2} + 235\cdot 337^{3} + 107\cdot 337^{4} + 67\cdot 337^{5} +O\left(337^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 139 + 8\cdot 337 + 217\cdot 337^{2} + 73\cdot 337^{4} + 166\cdot 337^{5} +O\left(337^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 168 + 19\cdot 337 + 23\cdot 337^{2} + 50\cdot 337^{3} + 104\cdot 337^{4} + 205\cdot 337^{5} +O\left(337^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 169 + 317\cdot 337 + 313\cdot 337^{2} + 286\cdot 337^{3} + 232\cdot 337^{4} + 131\cdot 337^{5} +O\left(337^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 198 + 328\cdot 337 + 119\cdot 337^{2} + 336\cdot 337^{3} + 263\cdot 337^{4} + 170\cdot 337^{5} +O\left(337^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 228 + 321\cdot 337 + 45\cdot 337^{2} + 101\cdot 337^{3} + 229\cdot 337^{4} + 269\cdot 337^{5} +O\left(337^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 237 + 174\cdot 337 + 94\cdot 337^{2} + 220\cdot 337^{3} + 124\cdot 337^{4} + 54\cdot 337^{5} +O\left(337^{ 6 }\right)$

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

Cycle notation

$(3,6)(4,5)$

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

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

$(2,7)(3,6)$

$(1,8)(3,6)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.