Galois orbit of Artin representations 4.2e8_3e4_7e2.8t15.1

• Label: 4.2e8_3e4_7e2.8t15.1
• Dimension: 4
• Group: $Z_8 : Z_8^\times$
• Conductor: $ 2^{8} \cdot 3^{4} \cdot 7^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$Z_8 : Z_8^\times$
Conductor:	$1016064= 2^{8} \cdot 3^{4} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{8} + 3 x^{6} - 6 x^{2} - 3 $ 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 $\Q_{ 421 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 2 + 230\cdot 421 + 265\cdot 421^{2} + 39\cdot 421^{3} + 167\cdot 421^{4} + 99\cdot 421^{5} + 260\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 137 + 238\cdot 421 + 232\cdot 421^{2} + 127\cdot 421^{3} + 188\cdot 421^{4} + 236\cdot 421^{5} + 10\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 153 + 36\cdot 421 + 377\cdot 421^{2} + 251\cdot 421^{3} + 329\cdot 421^{4} + 392\cdot 421^{5} + 166\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 206 + 34\cdot 421 + 50\cdot 421^{2} + 404\cdot 421^{3} + 33\cdot 421^{4} + 38\cdot 421^{5} + 166\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 215 + 386\cdot 421 + 370\cdot 421^{2} + 16\cdot 421^{3} + 387\cdot 421^{4} + 382\cdot 421^{5} + 254\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 268 + 384\cdot 421 + 43\cdot 421^{2} + 169\cdot 421^{3} + 91\cdot 421^{4} + 28\cdot 421^{5} + 254\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 284 + 182\cdot 421 + 188\cdot 421^{2} + 293\cdot 421^{3} + 232\cdot 421^{4} + 184\cdot 421^{5} + 410\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 419 + 190\cdot 421 + 155\cdot 421^{2} + 381\cdot 421^{3} + 253\cdot 421^{4} + 321\cdot 421^{5} + 160\cdot 421^{6} +O\left(421^{ 7 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.