Artin representation 2.2e5_17.4t3.2c1

• Label: 2.2e5_17.4t3.2c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 2^{5} \cdot 17 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$544= 2^{5} \cdot 17 $
Artin number field:	Splitting field of $f= x^{4} - 6 x^{2} - 4 x + 2 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Even
Determinant:	1.2e3_17.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 15 + 66\cdot 89 + 38\cdot 89^{2} + 73\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 29 + 2\cdot 89 + 72\cdot 89^{2} + 61\cdot 89^{3} + 8\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 49 + 88\cdot 89 + 10\cdot 89^{2} + 39\cdot 89^{3} + 79\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 85 + 20\cdot 89 + 56\cdot 89^{2} + 76\cdot 89^{3} + 16\cdot 89^{4} +O\left(89^{ 5 }\right)$

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

Cycle notation

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

$(2,4)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.