Artin representation 2.2e3_3e2_5.4t3.4c1

• Label: 2.2e3_3e2_5.4t3.4c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 2^{3} \cdot 3^{2} \cdot 5 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$360= 2^{3} \cdot 3^{2} \cdot 5 $
Artin number field:	Splitting field of $f= x^{4} - 3 x^{2} + 6 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.2e3_5.2t1.2c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 8 + 9\cdot 23 + 3\cdot 23^{2} + 22\cdot 23^{3} + 16\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 10 + 12\cdot 23 + 20\cdot 23^{2} + 14\cdot 23^{3} +O\left(23^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 13 + 10\cdot 23 + 2\cdot 23^{2} + 8\cdot 23^{3} + 22\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 15 + 13\cdot 23 + 19\cdot 23^{2} + 6\cdot 23^{4} +O\left(23^{ 5 }\right)$

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

Cycle notation

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

$(2,3)$

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.