Artin representation 2.13_113.4t3.1c1

• Label: 2.13_113.4t3.1c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 13 \cdot 113 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$1469= 13 \cdot 113 $
Artin number field:	Splitting field of $f= x^{4} - x^{3} + 7 x^{2} + 8 x + 12 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Even
Determinant:	1.13_113.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 11 + 12\cdot 53 + 32\cdot 53^{2} + 3\cdot 53^{3} + 28\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 20 + 5\cdot 53 + 31\cdot 53^{2} + 44\cdot 53^{3} + 19\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 26 + 40\cdot 53 + 35\cdot 53^{2} + 23\cdot 53^{3} + 28\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 50 + 47\cdot 53 + 6\cdot 53^{2} + 34\cdot 53^{3} + 29\cdot 53^{4} +O\left(53^{ 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 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.