Artin representation 2.2e4_5_31.4t3.6c1

• Label: 2.2e4_5_31.4t3.6c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 2^{4} \cdot 5 \cdot 31 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$2480= 2^{4} \cdot 5 \cdot 31 $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} - 8 x^{2} - 22 x + 121 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Even
Determinant:	1.2e2_5_31.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 64 + 11\cdot 109 + 90\cdot 109^{2} + 93\cdot 109^{3} + 77\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 70 + 81\cdot 109 + 6\cdot 109^{2} + 82\cdot 109^{3} + 47\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 94 + 2\cdot 109 + 36\cdot 109^{2} + 80\cdot 109^{3} + 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 101 + 12\cdot 109 + 85\cdot 109^{2} + 70\cdot 109^{3} + 90\cdot 109^{4} +O\left(109^{ 5 }\right)$

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

Cycle notation

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

$(3,4)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.