Galois orbit of Artin representations 4.2e12_3e2_5e2.8t22.11

• Label: 4.2e12_3e2_5e2.8t22.11
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{12} \cdot 3^{2} \cdot 5^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$921600= 2^{12} \cdot 3^{2} \cdot 5^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{6} + 2 x^{4} - 6 x^{2} + 9 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3 : D_4 $
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 241 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 12 + 10\cdot 241 + 80\cdot 241^{2} + 95\cdot 241^{3} + 94\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 43 + 194\cdot 241 + 168\cdot 241^{2} + 85\cdot 241^{3} + 39\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 85 + 66\cdot 241 + 234\cdot 241^{2} + 212\cdot 241^{3} + 220\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 119 + 95\cdot 241 + 77\cdot 241^{2} + 39\cdot 241^{3} + 114\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 122 + 145\cdot 241 + 163\cdot 241^{2} + 201\cdot 241^{3} + 126\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 156 + 174\cdot 241 + 6\cdot 241^{2} + 28\cdot 241^{3} + 20\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 198 + 46\cdot 241 + 72\cdot 241^{2} + 155\cdot 241^{3} + 201\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 229 + 230\cdot 241 + 160\cdot 241^{2} + 145\cdot 241^{3} + 146\cdot 241^{4} +O\left(241^{ 5 }\right)$

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

Cycle notation

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

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

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

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

$(1,8)(4,5)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.