Artin representation 2.11_37.4t3.2c1

• Label: 2.11_37.4t3.2c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 11 \cdot 37 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$407= 11 \cdot 37 $
Artin number field:	Splitting field of $f= x^{4} - x^{3} - 4 x + 5 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.11_37.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 3 + 36\cdot 47 + 39\cdot 47^{2} + 16\cdot 47^{3} + 16\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 24 + 32\cdot 47 + 16\cdot 47^{3} + 6\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 31 + 25\cdot 47 + 18\cdot 47^{2} + 29\cdot 47^{3} + 45\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 37 + 46\cdot 47 + 34\cdot 47^{2} + 31\cdot 47^{3} + 25\cdot 47^{4} +O\left(47^{ 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 value
$1$	$1$	$()$	$2$
$1$	$2$	$(1,2)(3,4)$	$-2$
$2$	$2$	$(1,3)(2,4)$	$0$
$2$	$2$	$(1,2)$	$0$
$2$	$4$	$(1,4,2,3)$	$0$

The blue line marks the conjugacy class containing complex conjugation.