Galois orbit of Artin representations 2.1205.4t3.a

• Label: 2.1205.4t3.a
• Dimension: $2$
• Group: $D_{4}$
• Conductor: $1205$
• Indicator: $1$

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	\(1205\)\(\medspace = 5 \cdot 241 \)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 4.0.290405.1
Galois orbit size:	$1$
Smallest permutation container:	$D_{4}$
Parity:	even
Projective image:	$C_2^2$
Projective field:	Galois closure of \(\Q(\sqrt{5}, \sqrt{241})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 8 + 8\cdot 41 + 11\cdot 41^{2} + 2\cdot 41^{3} + 8\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 12 + 32\cdot 41 + 7\cdot 41^{2} + 36\cdot 41^{3} + 9\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 30 + 8\cdot 41 + 33\cdot 41^{2} + 4\cdot 41^{3} + 31\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 34 + 32\cdot 41 + 29\cdot 41^{2} + 38\cdot 41^{3} + 32\cdot 41^{4} +O\left(41^{ 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.