Artin representation 2.3e2_7_61.4t3.4c1

• Label: 2.3e2_7_61.4t3.4c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 3^{2} \cdot 7 \cdot 61 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$3843= 3^{2} \cdot 7 \cdot 61 $
Artin number field:	Splitting field of $f= x^{4} - x^{3} + 18 x^{2} - 13 x + 79 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.7_61.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 72 + 139 + 115\cdot 139^{2} + 102\cdot 139^{3} + 119\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 110 + 42\cdot 139 + 135\cdot 139^{2} + 100\cdot 139^{3} + 6\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 116 + 129\cdot 139 + 135\cdot 139^{2} + 3\cdot 139^{3} + 89\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 120 + 103\cdot 139 + 30\cdot 139^{2} + 70\cdot 139^{3} + 62\cdot 139^{4} +O\left(139^{ 5 }\right)$

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

Cycle notation

$(1,2)$

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

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.