Galois orbit of Artin representations 2.5_191.4t3.2

• Label: 2.5_191.4t3.2
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 5 \cdot 191 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$955= 5 \cdot 191 $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} + 6 x^{2} - 5 x + 54 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 239 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 10 + 21\cdot 239 + 47\cdot 239^{2} + 209\cdot 239^{3} + 144\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 74 + 228\cdot 239 + 207\cdot 239^{2} + 180\cdot 239^{3} + 162\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 166 + 10\cdot 239 + 31\cdot 239^{2} + 58\cdot 239^{3} + 76\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 230 + 217\cdot 239 + 191\cdot 239^{2} + 29\cdot 239^{3} + 94\cdot 239^{4} +O\left(239^{ 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 values
			$c1$
$1$	$1$	$()$	$2$
$1$	$2$	$(1,4)(2,3)$	$-2$
$2$	$2$	$(1,2)(3,4)$	$0$
$2$	$2$	$(1,4)$	$0$
$2$	$4$	$(1,3,4,2)$	$0$

The blue line marks the conjugacy class containing complex conjugation.