Galois orbit of Artin representations 2.2e2_7_11.4t3.1

• Label: 2.2e2_7_11.4t3.1
• 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} + x^{2} - 11 $ 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 $\Q_{ 107 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 18 + 90\cdot 107 + 21\cdot 107^{2} + 68\cdot 107^{3} + 73\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 38 + 12\cdot 107 + 31\cdot 107^{2} + 92\cdot 107^{3} + 81\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 70 + 94\cdot 107 + 75\cdot 107^{2} + 14\cdot 107^{3} + 25\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 90 + 16\cdot 107 + 85\cdot 107^{2} + 38\cdot 107^{3} + 33\cdot 107^{4} +O\left(107^{ 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
			$c1$
$1$	$1$	$()$	$2$
$1$	$2$	$(1,4)(2,3)$	$-2$
$2$	$2$	$(1,2)(3,4)$	$0$
$2$	$2$	$(2,3)$	$0$
$2$	$4$	$(1,2,4,3)$	$0$

The blue line marks the conjugacy class containing complex conjugation.