Artin representation 2.2e2_193.4t3.1c1

• Label: 2.2e2_193.4t3.1c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 2^{2} \cdot 193 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$772= 2^{2} \cdot 193 $
Artin number field:	Splitting field of $f= x^{4} - 7 x^{2} - 36 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.2e2_193.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 197 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 65 + 57\cdot 197 + 25\cdot 197^{2} + 181\cdot 197^{3} + 64\cdot 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 71 + 132\cdot 197 + 158\cdot 197^{2} + 46\cdot 197^{3} + 195\cdot 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 126 + 64\cdot 197 + 38\cdot 197^{2} + 150\cdot 197^{3} + 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 132 + 139\cdot 197 + 171\cdot 197^{2} + 15\cdot 197^{3} + 132\cdot 197^{4} +O\left(197^{ 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.