Galois orbit of Artin representations 4.2e17_3e2.8t21.5

• Label: 4.2e17_3e2.8t21.5
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 2^{17} \cdot 3^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$1179648= 2^{17} \cdot 3^{2} $
Artin number field:	Splitting field of $f= x^{8} - 8 x^{5} + 36 x^{4} - 64 x^{3} + 64 x^{2} - 32 x + 6 $ 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 $\Q_{ 233 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 3 + 149\cdot 233 + 80\cdot 233^{2} + 56\cdot 233^{3} + 134\cdot 233^{4} + 149\cdot 233^{5} + 82\cdot 233^{6} +O\left(233^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 38 + 145\cdot 233 + 199\cdot 233^{2} + 191\cdot 233^{3} + 16\cdot 233^{4} + 223\cdot 233^{5} + 166\cdot 233^{6} +O\left(233^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 82 + 103\cdot 233 + 206\cdot 233^{2} + 56\cdot 233^{3} + 94\cdot 233^{4} + 119\cdot 233^{5} + 8\cdot 233^{6} +O\left(233^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 96 + 148\cdot 233 + 26\cdot 233^{2} + 121\cdot 233^{3} + 155\cdot 233^{4} + 51\cdot 233^{5} + 175\cdot 233^{6} +O\left(233^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 150 + 167\cdot 233 + 104\cdot 233^{2} + 114\cdot 233^{3} + 184\cdot 233^{4} + 85\cdot 233^{5} + 19\cdot 233^{6} +O\left(233^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 152 + 197\cdot 233 + 224\cdot 233^{2} + 160\cdot 233^{3} + 152\cdot 233^{4} + 174\cdot 233^{5} + 110\cdot 233^{6} +O\left(233^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 194 + 19\cdot 233 + 68\cdot 233^{2} + 56\cdot 233^{3} + 202\cdot 233^{4} + 181\cdot 233^{5} + 179\cdot 233^{6} +O\left(233^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 217 + 21\cdot 233^{2} + 174\cdot 233^{3} + 224\cdot 233^{4} + 178\cdot 233^{5} + 188\cdot 233^{6} +O\left(233^{ 7 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.