Artin representation 2.2e3_2113.4t3.3c1

• Label: 2.2e3_2113.4t3.3c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 2^{3} \cdot 2113 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$16904= 2^{3} \cdot 2113 $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} - 23 x^{2} + 24 x + 126 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Even
Determinant:	1.2e3_2113.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 191 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 28 + 64\cdot 191 + 85\cdot 191^{2} + 40\cdot 191^{3} + 142\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 40 + 52\cdot 191 + 67\cdot 191^{2} + 57\cdot 191^{3} + 78\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 152 + 138\cdot 191 + 123\cdot 191^{2} + 133\cdot 191^{3} + 112\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 164 + 126\cdot 191 + 105\cdot 191^{2} + 150\cdot 191^{3} + 48\cdot 191^{4} +O\left(191^{ 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.