Artin representation 2.3_7_19e2.4t3.4c1

• Label: 2.3_7_19e2.4t3.4c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 3 \cdot 7 \cdot 19^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$7581= 3 \cdot 7 \cdot 19^{2} $
Artin number field:	Splitting field of $f= x^{4} - x^{3} - 2 x^{2} + 51 x + 207 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Even
Determinant:	1.3_7.2t1.1c1

Galois action

Roots of defining polynomial

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

Roots:

$r_{ 1 }$	$=$	$ 13 + 8\cdot 41^{2} + 29\cdot 41^{3} + 13\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 16 + 27\cdot 41 + 3\cdot 41^{2} + 24\cdot 41^{3} +O\left(41^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 25 + 6\cdot 41 + 20\cdot 41^{2} + 9\cdot 41^{3} + 18\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 29 + 6\cdot 41 + 9\cdot 41^{2} + 19\cdot 41^{3} + 8\cdot 41^{4} +O\left(41^{ 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.