Artin representation 4.2e11_17e2.8t21.3c1

• Label: 4.2e11_17e2.8t21.3c1
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 2^{11} \cdot 17^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$591872= 2^{11} \cdot 17^{2} $
Artin number field:	Splitting field of $f= x^{8} + x^{6} + 2 x^{2} + 4 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3: C_4$
Parity:	Even
Determinant:	1.2e3.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 191 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 8 + 70\cdot 191 + 150\cdot 191^{2} + 148\cdot 191^{3} + 155\cdot 191^{4} + 159\cdot 191^{5} +O\left(191^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 14 + 150\cdot 191 + 23\cdot 191^{2} + 101\cdot 191^{3} + 21\cdot 191^{4} + 6\cdot 191^{5} +O\left(191^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 31 + 146\cdot 191 + 89\cdot 191^{2} + 68\cdot 191^{3} + 118\cdot 191^{4} + 3\cdot 191^{5} +O\left(191^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 45 + 29\cdot 191 + 113\cdot 191^{2} + 73\cdot 191^{3} + 56\cdot 191^{4} + 118\cdot 191^{5} +O\left(191^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 146 + 161\cdot 191 + 77\cdot 191^{2} + 117\cdot 191^{3} + 134\cdot 191^{4} + 72\cdot 191^{5} +O\left(191^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 160 + 44\cdot 191 + 101\cdot 191^{2} + 122\cdot 191^{3} + 72\cdot 191^{4} + 187\cdot 191^{5} +O\left(191^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 177 + 40\cdot 191 + 167\cdot 191^{2} + 89\cdot 191^{3} + 169\cdot 191^{4} + 184\cdot 191^{5} +O\left(191^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 183 + 120\cdot 191 + 40\cdot 191^{2} + 42\cdot 191^{3} + 35\cdot 191^{4} + 31\cdot 191^{5} +O\left(191^{ 6 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.