Galois orbit of Artin representations 4.5e4_131e2.8t16.1

• Label: 4.5e4_131e2.8t16.1
• Dimension: 4
• Group: $(C_8:C_2):C_2$
• Conductor: $ 5^{4} \cdot 131^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(C_8:C_2):C_2$
Conductor:	$10725625= 5^{4} \cdot 131^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} - 7 x^{6} - 4 x^{5} + 70 x^{4} + 41 x^{3} - 287 x^{2} - 62 x + 401 $ 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 $\Q_{ 271 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 90 + 160\cdot 271 + 188\cdot 271^{2} + 65\cdot 271^{3} + 264\cdot 271^{4} + 138\cdot 271^{5} +O\left(271^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 108 + 196\cdot 271 + 211\cdot 271^{2} + 101\cdot 271^{3} + 91\cdot 271^{4} + 248\cdot 271^{5} +O\left(271^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 114 + 171\cdot 271 + 50\cdot 271^{2} + 228\cdot 271^{3} + 48\cdot 271^{4} + 107\cdot 271^{5} +O\left(271^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 138 + 7\cdot 271 + 53\cdot 271^{2} + 180\cdot 271^{3} + 202\cdot 271^{4} + 87\cdot 271^{5} +O\left(271^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 190 + 9\cdot 271 + 77\cdot 271^{2} + 141\cdot 271^{3} + 117\cdot 271^{4} + 247\cdot 271^{5} +O\left(271^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 231 + 13\cdot 271 + 91\cdot 271^{2} + 146\cdot 271^{3} + 137\cdot 271^{4} + 47\cdot 271^{5} +O\left(271^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 242 + 40\cdot 271 + 245\cdot 271^{2} + 52\cdot 271^{3} + 122\cdot 271^{4} + 49\cdot 271^{5} +O\left(271^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 244 + 212\cdot 271 + 166\cdot 271^{2} + 167\cdot 271^{3} + 99\cdot 271^{4} + 157\cdot 271^{5} +O\left(271^{ 6 }\right)$

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

Cycle notation

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.