Galois orbit of Artin representations 2.2e7_7.4t3.1

• Label: 2.2e7_7.4t3.1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 2^{7} \cdot 7 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$896= 2^{7} \cdot 7 $
Artin number field:	Splitting field of $f= x^{4} - 6 x^{2} + 2 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 9 + 97\cdot 103 + 25\cdot 103^{2} + 19\cdot 103^{3} + 99\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 50 + 41\cdot 103 + 89\cdot 103^{2} + 41\cdot 103^{3} + 44\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 53 + 61\cdot 103 + 13\cdot 103^{2} + 61\cdot 103^{3} + 58\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 94 + 5\cdot 103 + 77\cdot 103^{2} + 83\cdot 103^{3} + 3\cdot 103^{4} +O\left(103^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

Cycle notation

$(1,4)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.