Artin representation 2.55.4t3.b.a

• Label: 2.55.4t3.b.a
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 5 \cdot 11 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$55= 5 \cdot 11 $
Artin number field:	Splitting field of 4.0.605.1 defined by $f= x^{4} - x^{3} + x^{2} + x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.55.2t1.a.a
Projective image:	$C_2^2$
Projective field:	Galois closure of \(\Q(\sqrt{5}, \sqrt{-11})\)

Galois action

Roots of defining polynomial

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

Roots:

$r_{ 1 }$	$=$	$ 5 + 15\cdot 59 + 32\cdot 59^{2} + 37\cdot 59^{3} + 21\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 32 + 47\cdot 59 + 19\cdot 59^{2} + 34\cdot 59^{3} + 20\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 35 + 7\cdot 59 + 38\cdot 59^{2} + 51\cdot 59^{3} + 44\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 47 + 47\cdot 59 + 27\cdot 59^{2} + 53\cdot 59^{3} + 30\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 value
$1$	$1$	$()$	$2$
$1$	$2$	$(1,4)(2,3)$	$-2$
$2$	$2$	$(1,2)(3,4)$	$0$
$2$	$2$	$(1,4)$	$0$
$2$	$4$	$(1,3,4,2)$	$0$

The blue line marks the conjugacy class containing complex conjugation.