Galois orbit of Artin representations 2.2e2_7_11.4t3.2

• Label: 2.2e2_7_11.4t3.2
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 2^{2} \cdot 7 \cdot 11 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$308= 2^{2} \cdot 7 \cdot 11 $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} - 5 x^{2} + 6 x - 2 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 11 + 27\cdot 43 + 17\cdot 43^{2} + 14\cdot 43^{3} + 16\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 16 + 30\cdot 43 + 43^{3} + 33\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 28 + 12\cdot 43 + 42\cdot 43^{2} + 41\cdot 43^{3} + 9\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 33 + 15\cdot 43 + 25\cdot 43^{2} + 28\cdot 43^{3} + 26\cdot 43^{4} +O\left(43^{ 5 }\right)$

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

Cycle notation

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

$(2,3)$

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.