Artin representation 2.3_11_359.4t3.3c1

• Label: 2.3_11_359.4t3.3c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 3 \cdot 11 \cdot 359 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$11847= 3 \cdot 11 \cdot 359 $
Artin number field:	Splitting field of $f= x^{4} - 13 x^{2} + 132 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.3_11_359.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 107 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 25 + 69\cdot 107 + 24\cdot 107^{2} + 46\cdot 107^{3} + 65\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 43 + 38\cdot 107 + 55\cdot 107^{2} + 59\cdot 107^{3} + 8\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 64 + 68\cdot 107 + 51\cdot 107^{2} + 47\cdot 107^{3} + 98\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 82 + 37\cdot 107 + 82\cdot 107^{2} + 60\cdot 107^{3} + 41\cdot 107^{4} +O\left(107^{ 5 }\right)$

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

Cycle notation

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

$(2,3)$

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.