Artin representation 2.2e4_3_13.4t3.3c1

• Label: 2.2e4_3_13.4t3.3c1
• 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} - 5 x^{2} + 3 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Even
Determinant:	1.2e2_3_13.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 107 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 21 + 42\cdot 107 + 107^{2} + 106\cdot 107^{3} + 62\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 45 + 17\cdot 107 + 81\cdot 107^{2} + 6\cdot 107^{3} + 3\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 62 + 89\cdot 107 + 25\cdot 107^{2} + 100\cdot 107^{3} + 103\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 86 + 64\cdot 107 + 105\cdot 107^{2} + 44\cdot 107^{4} +O\left(107^{ 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.