Artin representation 2.3_13e2_109.4t3.2c1

• Label: 2.3_13e2_109.4t3.2c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 3 \cdot 13^{2} \cdot 109 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$55263= 3 \cdot 13^{2} \cdot 109 $
Artin number field:	Splitting field of $f= x^{4} - x^{3} - 61 x^{2} + 55 x + 1153 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.3_109.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 1 + 17\cdot 31 + 9\cdot 31^{2} + 22\cdot 31^{3} + 24\cdot 31^{4} + 30\cdot 31^{5} +O\left(31^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 4 + 14\cdot 31 + 20\cdot 31^{2} + 16\cdot 31^{3} + 24\cdot 31^{4} + 19\cdot 31^{5} +O\left(31^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 5 + 28\cdot 31 + 14\cdot 31^{2} + 17\cdot 31^{3} + 24\cdot 31^{4} + 29\cdot 31^{5} +O\left(31^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 22 + 2\cdot 31 + 17\cdot 31^{2} + 5\cdot 31^{3} + 19\cdot 31^{4} + 12\cdot 31^{5} +O\left(31^{ 6 }\right)$

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

Cycle notation

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

$(1,3)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.