Galois orbit of Artin representations 2.384.4t3.b

• Label: 2.384.4t3.b
• Dimension: $2$
• Group: $D_{4}$
• Conductor: $384$
• Indicator: $1$

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 4.0.3072.2
Galois orbit size:	$1$
Smallest permutation container:	$D_{4}$
Parity:	odd
Projective image:	$C_2^2$
Projective field:	Galois closure of \(\Q(\sqrt{-2}, \sqrt{3})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 9 + 2\cdot 59 + 35\cdot 59^{2} + 26\cdot 59^{3} + 23\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 25 + 25\cdot 59 + 44\cdot 59^{2} + 17\cdot 59^{3} + 12\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 34 + 33\cdot 59 + 14\cdot 59^{2} + 41\cdot 59^{3} + 46\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 50 + 56\cdot 59 + 23\cdot 59^{2} + 32\cdot 59^{3} + 35\cdot 59^{4} +O\left(59^{ 5 }\right)$

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

Cycle notation

$(1,4)$

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

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.