Galois orbit of Artin representations 4.2e10_17e2.8t15.1

• Label: 4.2e10_17e2.8t15.1
• Dimension: 4
• Group: $Z_8 : Z_8^\times$
• Conductor: $ 2^{10} \cdot 17^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$Z_8 : Z_8^\times$
Conductor:	$295936= 2^{10} \cdot 17^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{6} + 5 x^{4} - 2 x^{2} - 4 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$Z_8 : Z_8^\times$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 409 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 65 + 269\cdot 409 + 269\cdot 409^{2} + 409^{3} + 86\cdot 409^{4} + 298\cdot 409^{5} + 188\cdot 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 72 + 237\cdot 409 + 400\cdot 409^{2} + 335\cdot 409^{3} + 144\cdot 409^{4} + 222\cdot 409^{5} + 253\cdot 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 104 + 407\cdot 409 + 408\cdot 409^{2} + 294\cdot 409^{3} + 139\cdot 409^{4} + 358\cdot 409^{5} + 295\cdot 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 135 + 24\cdot 409 + 172\cdot 409^{2} + 331\cdot 409^{3} + 41\cdot 409^{4} + 13\cdot 409^{5} + 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 274 + 384\cdot 409 + 236\cdot 409^{2} + 77\cdot 409^{3} + 367\cdot 409^{4} + 395\cdot 409^{5} + 407\cdot 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 305 + 409 + 114\cdot 409^{3} + 269\cdot 409^{4} + 50\cdot 409^{5} + 113\cdot 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 337 + 171\cdot 409 + 8\cdot 409^{2} + 73\cdot 409^{3} + 264\cdot 409^{4} + 186\cdot 409^{5} + 155\cdot 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 344 + 139\cdot 409 + 139\cdot 409^{2} + 407\cdot 409^{3} + 322\cdot 409^{4} + 110\cdot 409^{5} + 220\cdot 409^{6} +O\left(409^{ 7 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.