Artin representation 2.5_31_41.4t3.12c1

• Label: 2.5_31_41.4t3.12c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 5 \cdot 31 \cdot 41 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$6355= 5 \cdot 31 \cdot 41 $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} - 8 x^{2} + 9 x + 338 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.5_31_41.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 19 + 17\cdot 59 + 47\cdot 59^{2} + 45\cdot 59^{3} + 10\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 24 + 34\cdot 59 + 3\cdot 59^{2} + 38\cdot 59^{3} + 35\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 36 + 24\cdot 59 + 55\cdot 59^{2} + 20\cdot 59^{3} + 23\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 41 + 41\cdot 59 + 11\cdot 59^{2} + 13\cdot 59^{3} + 48\cdot 59^{4} +O\left(59^{ 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.