Artin representation 2.3_23e2.4t3.2c1

• Label: 2.3_23e2.4t3.2c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 3 \cdot 23^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$1587= 3 \cdot 23^{2} $
Artin number field:	Splitting field of $f= x^{4} - x^{3} + 9 x^{2} + 10 x + 8 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.3.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 10 + 36\cdot 139 + 118\cdot 139^{2} + 69\cdot 139^{3} + 19\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 58 + 32\cdot 139 + 83\cdot 139^{2} + 17\cdot 139^{3} + 104\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 98 + 127\cdot 139 + 12\cdot 139^{2} + 58\cdot 139^{3} + 59\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 113 + 81\cdot 139 + 63\cdot 139^{2} + 132\cdot 139^{3} + 94\cdot 139^{4} +O\left(139^{ 5 }\right)$

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

Cycle notation

$(1,4)$

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

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.