Galois orbit of Artin representations 4.2e14_17e2.8t15.2

• Label: 4.2e14_17e2.8t15.2
• Dimension: 4
• Group: $Z_8 : Z_8^\times$
• Conductor: $ 2^{14} \cdot 17^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$Z_8 : Z_8^\times$
Conductor:	$4734976= 2^{14} \cdot 17^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{6} + 10 x^{4} - 4 x^{2} + 68 $ 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 $\Q_{ 353 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 31 + 120\cdot 353 + 246\cdot 353^{3} + 191\cdot 353^{4} + 336\cdot 353^{5} + 264\cdot 353^{6} + 94\cdot 353^{7} +O\left(353^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 91 + 320\cdot 353 + 98\cdot 353^{2} + 308\cdot 353^{3} + 134\cdot 353^{4} + 194\cdot 353^{5} + 201\cdot 353^{6} + 155\cdot 353^{7} +O\left(353^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 130 + 333\cdot 353 + 26\cdot 353^{2} + 163\cdot 353^{3} + 12\cdot 353^{4} + 169\cdot 353^{5} + 340\cdot 353^{6} + 164\cdot 353^{7} +O\left(353^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 138 + 292\cdot 353 + 193\cdot 353^{2} + 79\cdot 353^{3} + 58\cdot 353^{4} + 25\cdot 353^{5} + 186\cdot 353^{6} + 236\cdot 353^{7} +O\left(353^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 215 + 60\cdot 353 + 159\cdot 353^{2} + 273\cdot 353^{3} + 294\cdot 353^{4} + 327\cdot 353^{5} + 166\cdot 353^{6} + 116\cdot 353^{7} +O\left(353^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 223 + 19\cdot 353 + 326\cdot 353^{2} + 189\cdot 353^{3} + 340\cdot 353^{4} + 183\cdot 353^{5} + 12\cdot 353^{6} + 188\cdot 353^{7} +O\left(353^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 262 + 32\cdot 353 + 254\cdot 353^{2} + 44\cdot 353^{3} + 218\cdot 353^{4} + 158\cdot 353^{5} + 151\cdot 353^{6} + 197\cdot 353^{7} +O\left(353^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 322 + 232\cdot 353 + 352\cdot 353^{2} + 106\cdot 353^{3} + 161\cdot 353^{4} + 16\cdot 353^{5} + 88\cdot 353^{6} + 258\cdot 353^{7} +O\left(353^{ 8 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.