Galois orbit of Artin representations 4.2e10_3e2_5e2.8t22.1

• Label: 4.2e10_3e2_5e2.8t22.1
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{10} \cdot 3^{2} \cdot 5^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$230400= 2^{10} \cdot 3^{2} \cdot 5^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{7} + 6 x^{6} - 4 x^{5} - 4 x^{4} + 4 x^{3} + 4 x^{2} - 2 $ 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:

$r_{ 1 }$	$=$	$ 44 + 30\cdot 241 + 35\cdot 241^{2} + 87\cdot 241^{3} + 139\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 61 + 18\cdot 241 + 222\cdot 241^{2} + 181\cdot 241^{3} + 18\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 80 + 62\cdot 241 + 199\cdot 241^{2} + 219\cdot 241^{3} + 16\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 85 + 63\cdot 241 + 201\cdot 241^{2} + 27\cdot 241^{3} + 230\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 167 + 159\cdot 241 + 62\cdot 241^{2} + 179\cdot 241^{3} + 194\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 171 + 240\cdot 241 + 236\cdot 241^{2} + 92\cdot 241^{3} + 38\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 177 + 86\cdot 241 + 145\cdot 241^{2} + 216\cdot 241^{3} + 135\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 183 + 61\cdot 241 + 102\cdot 241^{2} + 199\cdot 241^{3} + 189\cdot 241^{4} +O\left(241^{ 5 }\right)$

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

Cycle notation

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

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

$(1,3)(2,4)$

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.