Artin representation 2.2e4_3_13.4t3.6c1

• Label: 2.2e4_3_13.4t3.6c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 2^{4} \cdot 3 \cdot 13 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$624= 2^{4} \cdot 3 \cdot 13 $
Artin number field:	Splitting field of $f= x^{4} + 7 x^{2} + 13 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.3_13.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 17 + 47\cdot 79 + 52\cdot 79^{2} + 18\cdot 79^{3} + 32\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 39 + 41\cdot 79 + 31\cdot 79^{2} + 8\cdot 79^{3} + 23\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 40 + 37\cdot 79 + 47\cdot 79^{2} + 70\cdot 79^{3} + 55\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 62 + 31\cdot 79 + 26\cdot 79^{2} + 60\cdot 79^{3} + 46\cdot 79^{4} +O\left(79^{ 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.