Artin representation 2.2e2_3_7_19.4t3.6c1

• Label: 2.2e2_3_7_19.4t3.6c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 2^{2} \cdot 3 \cdot 7 \cdot 19 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$1596= 2^{2} \cdot 3 \cdot 7 \cdot 19 $
Artin number field:	Splitting field of $f= x^{4} + 3 x^{2} - 14 x + 18 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Even
Determinant:	1.2e2_3_7_19.2t1.1c1

Galois action

Roots of defining polynomial

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

Roots:

$r_{ 1 }$	$=$	$ 11 + 23\cdot 59 + 59^{2} + 6\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 29 + 59 + 29\cdot 59^{2} + 47\cdot 59^{3} + 53\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 38 + 51\cdot 59 + 19\cdot 59^{2} + 20\cdot 59^{3} + 14\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 40 + 41\cdot 59 + 8\cdot 59^{2} + 50\cdot 59^{3} + 43\cdot 59^{4} +O\left(59^{ 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.