Galois orbit of Artin representations 2.5_19_29.4t3.1

• Label: 2.5_19_29.4t3.1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 5 \cdot 19 \cdot 29 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$2755= 5 \cdot 19 \cdot 29 $
Artin number field:	Splitting field of $f= x^{4} - x^{3} + 5 x^{2} + 12 x - 36 $ 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 $\Q_{ 41 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 23 + 23\cdot 41 + 21\cdot 41^{3} + 4\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 25 + 39\cdot 41 + 18\cdot 41^{2} + 30\cdot 41^{3} + 23\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 36 + 10\cdot 41 + 6\cdot 41^{2} + 30\cdot 41^{3} + 12\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 40 + 7\cdot 41 + 15\cdot 41^{2} +O\left(41^{ 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.