Galois orbit of Artin representations 4.2e14_3e2_5e2.8t22.10

• Label: 4.2e14_3e2_5e2.8t22.10
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{14} \cdot 3^{2} \cdot 5^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$3686400= 2^{14} \cdot 3^{2} \cdot 5^{2} $
Artin number field:	Splitting field of $f= x^{8} - 10 x^{6} + 62 x^{4} - 140 x^{2} + 100 $ 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 6.

Roots:

$r_{ 1 }$	$=$	$ 3 + 153\cdot 241 + 178\cdot 241^{2} + 168\cdot 241^{3} + 47\cdot 241^{4} + 181\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 12 + 170\cdot 241 + 62\cdot 241^{2} + 87\cdot 241^{3} + 228\cdot 241^{4} + 60\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 53 + 177\cdot 241 + 39\cdot 241^{2} + 103\cdot 241^{3} + 223\cdot 241^{4} + 121\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 120 + 214\cdot 241 + 30\cdot 241^{2} + 160\cdot 241^{3} + 142\cdot 241^{4} + 202\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 121 + 26\cdot 241 + 210\cdot 241^{2} + 80\cdot 241^{3} + 98\cdot 241^{4} + 38\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 188 + 63\cdot 241 + 201\cdot 241^{2} + 137\cdot 241^{3} + 17\cdot 241^{4} + 119\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 229 + 70\cdot 241 + 178\cdot 241^{2} + 153\cdot 241^{3} + 12\cdot 241^{4} + 180\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 238 + 87\cdot 241 + 62\cdot 241^{2} + 72\cdot 241^{3} + 193\cdot 241^{4} + 59\cdot 241^{5} +O\left(241^{ 6 }\right)$

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

Cycle notation

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

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

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.