Galois orbit of Artin representations 4.2e11_3e2_7e2.8t21.1

• Label: 4.2e11_3e2_7e2.8t21.1
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 2^{11} \cdot 3^{2} \cdot 7^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$903168= 2^{11} \cdot 3^{2} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{6} + 8 x^{4} - 6 x^{2} + 4 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3: C_4$
Parity:	Even

Galois action

Roots of defining polynomial

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

Roots:

$r_{ 1 }$	$=$	$ 18 + 6\cdot 191 + 135\cdot 191^{2} + 74\cdot 191^{3} + 174\cdot 191^{4} + 6\cdot 191^{5} + 95\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 23 + 53\cdot 191 + 166\cdot 191^{2} + 130\cdot 191^{3} + 111\cdot 191^{4} + 29\cdot 191^{5} + 103\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 35 + 110\cdot 191 + 153\cdot 191^{2} + 98\cdot 191^{3} + 124\cdot 191^{4} + 175\cdot 191^{5} + 89\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 44 + 110\cdot 191 + 131\cdot 191^{2} + 168\cdot 191^{3} + 34\cdot 191^{4} + 72\cdot 191^{5} + 99\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 147 + 80\cdot 191 + 59\cdot 191^{2} + 22\cdot 191^{3} + 156\cdot 191^{4} + 118\cdot 191^{5} + 91\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 156 + 80\cdot 191 + 37\cdot 191^{2} + 92\cdot 191^{3} + 66\cdot 191^{4} + 15\cdot 191^{5} + 101\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 168 + 137\cdot 191 + 24\cdot 191^{2} + 60\cdot 191^{3} + 79\cdot 191^{4} + 161\cdot 191^{5} + 87\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 173 + 184\cdot 191 + 55\cdot 191^{2} + 116\cdot 191^{3} + 16\cdot 191^{4} + 184\cdot 191^{5} + 95\cdot 191^{6} +O\left(191^{ 7 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.