Artin representation 2.2e3_3e2_7.4t3.2c1

• Label: 2.2e3_3e2_7.4t3.2c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 2^{3} \cdot 3^{2} \cdot 7 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$504= 2^{3} \cdot 3^{2} \cdot 7 $
Artin number field:	Splitting field of $f= x^{4} - x^{3} + 5 x^{2} - 5 x + 4 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.2e3_7.2t1.2c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 7 + 21\cdot 71 + 23\cdot 71^{2} + 27\cdot 71^{3} + 27\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 18 + 6\cdot 71 + 12\cdot 71^{2} + 15\cdot 71^{3} + 65\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 22 + 9\cdot 71 + 11\cdot 71^{2} + 46\cdot 71^{3} + 12\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 25 + 34\cdot 71 + 24\cdot 71^{2} + 53\cdot 71^{3} + 36\cdot 71^{4} +O\left(71^{ 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.