Artin representation 4.5e4_11e2_41e2.8t16.6c1

• Label: 4.5e4_11e2_41e2.8t16.6c1
• Dimension: 4
• Group: $(C_8:C_2):C_2$
• Conductor: $ 5^{4} \cdot 11^{2} \cdot 41^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(C_8:C_2):C_2$
Conductor:	$127125625= 5^{4} \cdot 11^{2} \cdot 41^{2} $
Artin number field:	Splitting field of $f= x^{8} - x^{7} - 33 x^{6} + 37 x^{5} + 395 x^{4} - 382 x^{3} - 2018 x^{2} + 1081 x + 4051 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(C_8:C_2):C_2$
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 191 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 34 + 60\cdot 191 + 115\cdot 191^{2} + 44\cdot 191^{3} + 179\cdot 191^{4} + 69\cdot 191^{5} + 86\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 68 + 34\cdot 191 + 153\cdot 191^{2} + 114\cdot 191^{4} + 23\cdot 191^{5} + 112\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 84 + 47\cdot 191 + 149\cdot 191^{2} + 99\cdot 191^{3} + 104\cdot 191^{4} + 5\cdot 191^{5} + 50\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 89 + 101\cdot 191 + 86\cdot 191^{2} + 27\cdot 191^{3} + 89\cdot 191^{4} + 93\cdot 191^{5} + 120\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 108 + 86\cdot 191 + 68\cdot 191^{2} + 129\cdot 191^{3} + 155\cdot 191^{4} + 132\cdot 191^{5} + 67\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 109 + 174\cdot 191 + 161\cdot 191^{2} + 83\cdot 191^{3} + 172\cdot 191^{4} + 39\cdot 191^{5} + 41\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 114 + 184\cdot 191 + 97\cdot 191^{2} + 44\cdot 191^{3} + 133\cdot 191^{4} + 13\cdot 191^{5} + 132\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 159 + 74\cdot 191 + 122\cdot 191^{2} + 142\cdot 191^{3} + 6\cdot 191^{4} + 3\cdot 191^{5} + 154\cdot 191^{6} +O\left(191^{ 7 }\right)$

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

Cycle notation

$(2,3)(7,8)$

$(4,6)(7,8)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.