Galois orbit of Artin representations 4.2e8_5e4.8t16.2

• Label: 4.2e8_5e4.8t16.2
• Dimension: 4
• Group: $(C_8:C_2):C_2$
• Conductor: $ 2^{8} \cdot 5^{4}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(C_8:C_2):C_2$
Conductor:	$160000= 2^{8} \cdot 5^{4} $
Artin number field:	Splitting field of $f= x^{8} - 5 x^{6} + 10 x^{4} - 10 x^{2} + 5 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(C_8:C_2):C_2$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 461 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 36 + 146\cdot 461 + 416\cdot 461^{2} + 237\cdot 461^{3} + 154\cdot 461^{4} + 122\cdot 461^{5} +O\left(461^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 118 + 151\cdot 461 + 194\cdot 461^{2} + 37\cdot 461^{3} + 29\cdot 461^{4} + 224\cdot 461^{5} +O\left(461^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 185 + 235\cdot 461 + 149\cdot 461^{2} + 342\cdot 461^{3} + 454\cdot 461^{4} + 220\cdot 461^{5} +O\left(461^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 229 + 30\cdot 461 + 309\cdot 461^{2} + 294\cdot 461^{3} + 231\cdot 461^{4} + 305\cdot 461^{5} +O\left(461^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 232 + 430\cdot 461 + 151\cdot 461^{2} + 166\cdot 461^{3} + 229\cdot 461^{4} + 155\cdot 461^{5} +O\left(461^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 276 + 225\cdot 461 + 311\cdot 461^{2} + 118\cdot 461^{3} + 6\cdot 461^{4} + 240\cdot 461^{5} +O\left(461^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 343 + 309\cdot 461 + 266\cdot 461^{2} + 423\cdot 461^{3} + 431\cdot 461^{4} + 236\cdot 461^{5} +O\left(461^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 425 + 314\cdot 461 + 44\cdot 461^{2} + 223\cdot 461^{3} + 306\cdot 461^{4} + 338\cdot 461^{5} +O\left(461^{ 6 }\right)$

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

Cycle notation

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

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

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

$(1,8)(3,6)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.