Artin representation 2.3_13_109.4t3.3c1

• Label: 2.3_13_109.4t3.3c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 3 \cdot 13 \cdot 109 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$4251= 3 \cdot 13 \cdot 109 $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} - 17 x^{2} + 18 x + 84 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.3_13_109.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 6 + 24\cdot 37 + 9\cdot 37^{2} + 11\cdot 37^{3} + 14\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 15 + 6\cdot 37 + 21\cdot 37^{2} + 8\cdot 37^{3} + 32\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 23 + 30\cdot 37 + 15\cdot 37^{2} + 28\cdot 37^{3} + 4\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 32 + 12\cdot 37 + 27\cdot 37^{2} + 25\cdot 37^{3} + 22\cdot 37^{4} +O\left(37^{ 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 value
$1$	$1$	$()$	$2$
$1$	$2$	$(1,4)(2,3)$	$-2$
$2$	$2$	$(1,2)(3,4)$	$0$
$2$	$2$	$(1,4)$	$0$
$2$	$4$	$(1,3,4,2)$	$0$

The blue line marks the conjugacy class containing complex conjugation.