Artin representation 2.2e3_3e2_17.4t3.4c1

• Label: 2.2e3_3e2_17.4t3.4c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 2^{3} \cdot 3^{2} \cdot 17 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$1224= 2^{3} \cdot 3^{2} \cdot 17 $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} - 7 x^{2} + 2 x + 7 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Even
Determinant:	1.2e3_17.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 12 + 135\cdot 137 + 34\cdot 137^{2} + 80\cdot 137^{3} +O\left(137^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 19 + 36\cdot 137 + 121\cdot 137^{2} + 82\cdot 137^{3} + 56\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 20 + 119\cdot 137 + 119\cdot 137^{2} + 105\cdot 137^{3} + 45\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 88 + 120\cdot 137 + 134\cdot 137^{2} + 4\cdot 137^{3} + 34\cdot 137^{4} +O\left(137^{ 5 }\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.