Galois orbit of Artin representations 2.3_7_127.4t3.4

• Label: 2.3_7_127.4t3.4
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 3 \cdot 7 \cdot 127 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$2667= 3 \cdot 7 \cdot 127 $
Artin number field:	Splitting field of $f= x^{4} - x^{3} - 14 x^{2} + 9 x + 57 $ 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 $\Q_{ 223 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 19 + 14\cdot 223 + 168\cdot 223^{2} + 145\cdot 223^{3} + 138\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 21 + 96\cdot 223 + 82\cdot 223^{2} + 223^{3} + 86\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 56 + 106\cdot 223 + 189\cdot 223^{2} + 91\cdot 223^{3} + 128\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 128 + 6\cdot 223 + 6\cdot 223^{2} + 207\cdot 223^{3} + 92\cdot 223^{4} +O\left(223^{ 5 }\right)$

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

Cycle notation

$(1,2)$

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 4 }$	Character values
			$c1$
$1$	$1$	$()$	$2$
$1$	$2$	$(1,2)(3,4)$	$-2$
$2$	$2$	$(1,3)(2,4)$	$0$
$2$	$2$	$(1,2)$	$0$
$2$	$4$	$(1,4,2,3)$	$0$

The blue line marks the conjugacy class containing complex conjugation.