Artin representation 2.3_7_13_19.4t3.6c1

• Label: 2.3_7_13_19.4t3.6c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 3 \cdot 7 \cdot 13 \cdot 19 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$5187= 3 \cdot 7 \cdot 13 \cdot 19 $
Artin number field:	Splitting field of $f= x^{4} - x^{3} - 17 x^{2} + 18 x + 108 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.3_7_13_19.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 12 + 36\cdot 67 + 22\cdot 67^{2} + 10\cdot 67^{3} + 7\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 18 + 41\cdot 67 + 66\cdot 67^{2} + 55\cdot 67^{3} + 34\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 46 + 39\cdot 67 + 4\cdot 67^{2} + 52\cdot 67^{3} + 44\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 59 + 16\cdot 67 + 40\cdot 67^{2} + 15\cdot 67^{3} + 47\cdot 67^{4} +O\left(67^{ 5 }\right)$

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

Cycle notation

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

$(3,4)$

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.