Artin representation 2.37_41.4t3.1c1

• Label: 2.37_41.4t3.1c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 37 \cdot 41 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$1517= 37 \cdot 41 $
Artin number field:	Splitting field of $f= x^{4} - x^{3} + 3 x^{2} + 14 x + 32 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Even
Determinant:	1.37_41.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 24 + 35\cdot 83 + 22\cdot 83^{2} + 23\cdot 83^{3} + 45\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 38 + 42\cdot 83 + 78\cdot 83^{2} + 63\cdot 83^{3} + 46\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 41 + 44\cdot 83^{2} + 40\cdot 83^{3} + 66\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 64 + 4\cdot 83 + 21\cdot 83^{2} + 38\cdot 83^{3} + 7\cdot 83^{4} +O\left(83^{ 5 }\right)$

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

Cycle notation

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

$(2,4)$

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.