Artin representation 2.2e2_3_71.4t3.3c1

• Label: 2.2e2_3_71.4t3.3c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 2^{2} \cdot 3 \cdot 71 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$852= 2^{2} \cdot 3 \cdot 71 $
Artin number field:	Splitting field of $f= x^{4} - x^{3} + 12 x^{2} + 3 x + 9 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.2e2_3_71.2t1.1c1

Galois action

Roots of defining polynomial

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

Roots:

$r_{ 1 }$	$=$	$ 24 + 29\cdot 83 + 52\cdot 83^{2} + 38\cdot 83^{3} + 66\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 31 + 59\cdot 83 + 16\cdot 83^{2} + 82\cdot 83^{3} + 75\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 36 + 20\cdot 83 + 36\cdot 83^{2} + 67\cdot 83^{3} + 42\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 76 + 56\cdot 83 + 60\cdot 83^{2} + 60\cdot 83^{3} + 63\cdot 83^{4} +O\left(83^{ 5 }\right)$

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

Cycle notation

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

$(3,4)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 4 }$	Character value
$1$	$1$	$()$	$2$
$1$	$2$	$(1,2)(3,4)$	$-2$
$2$	$2$	$(1,3)(2,4)$	$0$
$2$	$2$	$(1,2)$	$0$
$2$	$4$	$(1,4,2,3)$	$0$

The blue line marks the conjugacy class containing complex conjugation.