Galois orbit of Artin representations 4.2e8_3e4_7e2.8t15.2

• Label: 4.2e8_3e4_7e2.8t15.2
• 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:

$r_{ 1 }$	$=$	$ 58 + 413\cdot 421 + 143\cdot 421^{2} + 55\cdot 421^{3} + 269\cdot 421^{4} + 139\cdot 421^{5} + 377\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 80 + 238\cdot 421 + 46\cdot 421^{2} + 137\cdot 421^{3} + 51\cdot 421^{4} + 98\cdot 421^{5} + 194\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 184 + 359\cdot 421 + 38\cdot 421^{2} + 10\cdot 421^{3} + 312\cdot 421^{4} + 53\cdot 421^{5} + 154\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 194 + 402\cdot 421 + 88\cdot 421^{2} + 239\cdot 421^{3} + 204\cdot 421^{4} + 29\cdot 421^{5} + 230\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 227 + 18\cdot 421 + 332\cdot 421^{2} + 181\cdot 421^{3} + 216\cdot 421^{4} + 391\cdot 421^{5} + 190\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 237 + 61\cdot 421 + 382\cdot 421^{2} + 410\cdot 421^{3} + 108\cdot 421^{4} + 367\cdot 421^{5} + 266\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 341 + 182\cdot 421 + 374\cdot 421^{2} + 283\cdot 421^{3} + 369\cdot 421^{4} + 322\cdot 421^{5} + 226\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 363 + 7\cdot 421 + 277\cdot 421^{2} + 365\cdot 421^{3} + 151\cdot 421^{4} + 281\cdot 421^{5} + 43\cdot 421^{6} +O\left(421^{ 7 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.