Artin representation 2.5_11_19.4t3.1c1

• Label: 2.5_11_19.4t3.1c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 5 \cdot 11 \cdot 19 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$1045= 5 \cdot 11 \cdot 19 $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} + 16 x^{2} - 15 x + 4 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Even
Determinant:	1.5_11_19.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 10 + 29 + 4\cdot 29^{2} + 13\cdot 29^{3} + 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 13 + 7\cdot 29 + 26\cdot 29^{2} + 2\cdot 29^{3} + 16\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 17 + 21\cdot 29 + 2\cdot 29^{2} + 26\cdot 29^{3} + 12\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 20 + 27\cdot 29 + 24\cdot 29^{2} + 15\cdot 29^{3} + 27\cdot 29^{4} +O\left(29^{ 5 }\right)$

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

Cycle notation

$(1,4)$

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

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.