Galois orbit of Artin representations 4.5e4_211e2.8t16.1

• Label: 4.5e4_211e2.8t16.1
• Dimension: 4
• Group: $(C_8:C_2):C_2$
• Conductor: $ 5^{4} \cdot 211^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(C_8:C_2):C_2$
Conductor:	$27825625= 5^{4} \cdot 211^{2} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} + 23 x^{6} - 36 x^{5} + 185 x^{4} - 141 x^{3} + 633 x^{2} - 188 x + 781 $ 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 $\Q_{ 241 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 31 + 94\cdot 241 + 136\cdot 241^{2} + 69\cdot 241^{3} + 5\cdot 241^{4} + 45\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 49 + 181\cdot 241 + 52\cdot 241^{2} + 179\cdot 241^{3} + 222\cdot 241^{4} + 151\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 57 + 132\cdot 241 + 59\cdot 241^{2} + 207\cdot 241^{3} + 221\cdot 241^{4} + 159\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 157 + 233\cdot 241 + 146\cdot 241^{2} + 73\cdot 241^{3} + 209\cdot 241^{4} + 30\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 162 + 147\cdot 241 + 62\cdot 241^{2} + 11\cdot 241^{3} + 196\cdot 241^{4} + 220\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 167 + 12\cdot 241 + 154\cdot 241^{2} + 172\cdot 241^{3} + 163\cdot 241^{4} + 209\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 171 + 102\cdot 241 + 168\cdot 241^{2} + 70\cdot 241^{3} + 122\cdot 241^{4} + 46\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 173 + 59\cdot 241 + 183\cdot 241^{2} + 179\cdot 241^{3} + 63\cdot 241^{4} + 99\cdot 241^{5} +O\left(241^{ 6 }\right)$

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

Cycle notation

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

$(5,7)(6,8)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.