Artin representation 2.2e2_3_11_59.4t3.5c1

• Label: 2.2e2_3_11_59.4t3.5c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 2^{2} \cdot 3 \cdot 11 \cdot 59 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$7788= 2^{2} \cdot 3 \cdot 11 \cdot 59 $
Artin number field:	Splitting field of $f= x^{4} + 29 x^{2} + 48 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Even
Determinant:	1.2e2_3_11_59.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 1 + 2\cdot 13 + 11\cdot 13^{2} + 8\cdot 13^{3} + 6\cdot 13^{4} + 11\cdot 13^{5} + 12\cdot 13^{6} + 6\cdot 13^{7} +O\left(13^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 3 + 13 + 4\cdot 13^{2} + 13^{3} + 13^{4} + 12\cdot 13^{6} + 7\cdot 13^{7} +O\left(13^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 10 + 11\cdot 13 + 8\cdot 13^{2} + 11\cdot 13^{3} + 11\cdot 13^{4} + 12\cdot 13^{5} + 5\cdot 13^{7} +O\left(13^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 12 + 10\cdot 13 + 13^{2} + 4\cdot 13^{3} + 6\cdot 13^{4} + 13^{5} + 6\cdot 13^{7} +O\left(13^{ 8 }\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.