Galois orbit of Artin representations 4.2e8_7e4.8t15.2

• Label: 4.2e8_7e4.8t15.2
• Dimension: 4
• Group: $Z_8 : Z_8^\times$
• Conductor: $ 2^{8} \cdot 7^{4}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$Z_8 : Z_8^\times$
Conductor:	$614656= 2^{8} \cdot 7^{4} $
Artin number field:	Splitting field of $f= x^{8} + 7 x^{6} + 21 x^{4} + 21 x^{2} + 14 $ 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 $\Q_{ 337 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 14 + 335\cdot 337 + 162\cdot 337^{2} + 117\cdot 337^{3} + 63\cdot 337^{4} + 57\cdot 337^{5} + 308\cdot 337^{6} + 116\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 63 + 157\cdot 337 + 48\cdot 337^{2} + 294\cdot 337^{3} + 18\cdot 337^{4} + 164\cdot 337^{5} + 5\cdot 337^{6} + 41\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 66 + 170\cdot 337 + 231\cdot 337^{2} + 42\cdot 337^{3} + 64\cdot 337^{4} + 294\cdot 337^{5} + 101\cdot 337^{6} + 273\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 137 + 248\cdot 337 + 34\cdot 337^{2} + 104\cdot 337^{3} + 54\cdot 337^{4} + 188\cdot 337^{5} + 176\cdot 337^{6} + 216\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 200 + 88\cdot 337 + 302\cdot 337^{2} + 232\cdot 337^{3} + 282\cdot 337^{4} + 148\cdot 337^{5} + 160\cdot 337^{6} + 120\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 271 + 166\cdot 337 + 105\cdot 337^{2} + 294\cdot 337^{3} + 272\cdot 337^{4} + 42\cdot 337^{5} + 235\cdot 337^{6} + 63\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 274 + 179\cdot 337 + 288\cdot 337^{2} + 42\cdot 337^{3} + 318\cdot 337^{4} + 172\cdot 337^{5} + 331\cdot 337^{6} + 295\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 323 + 337 + 174\cdot 337^{2} + 219\cdot 337^{3} + 273\cdot 337^{4} + 279\cdot 337^{5} + 28\cdot 337^{6} + 220\cdot 337^{7} +O\left(337^{ 8 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.