Artin representation 2.5_2039.4t3.2c1

• Label: 2.5_2039.4t3.2c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 5 \cdot 2039 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$10195= 5 \cdot 2039 $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} + 18 x^{2} - 17 x + 582 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.5_2039.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 2 + 11\cdot 31 + 5\cdot 31^{2} + 25\cdot 31^{3} + 14\cdot 31^{4} + 9\cdot 31^{5} +O\left(31^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 4 + 4\cdot 31 + 9\cdot 31^{2} + 11\cdot 31^{3} + 19\cdot 31^{4} + 26\cdot 31^{5} +O\left(31^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 28 + 26\cdot 31 + 21\cdot 31^{2} + 19\cdot 31^{3} + 11\cdot 31^{4} + 4\cdot 31^{5} +O\left(31^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 30 + 19\cdot 31 + 25\cdot 31^{2} + 5\cdot 31^{3} + 16\cdot 31^{4} + 21\cdot 31^{5} +O\left(31^{ 6 }\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.