Artin representation 2.1164.4t3.a.a

• Label: 2.1164.4t3.a.a
• Dimension: $2$
• Group: $D_{4}$
• Conductor: $1164$
• Root number: $1$
• Indicator: $1$

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$1164$$\medspace = 2^{2} \cdot 3 \cdot 97$
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin stem field:	Galois closure of 4.4.112908.1
Galois orbit size:	$1$
Smallest permutation container:	$D_{4}$
Parity:	even
Determinant:	1.1164.2t1.a.a
Projective image:	$C_2^2$
Projective field:	Galois closure of $\Q(\sqrt{3}, \sqrt{97})$

Defining polynomial

$f(x)$

$=$

$x^{4} - x^{3} - 16x^{2} - 4x + 16$ .

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$11 + 6\cdot 61 + 2\cdot 61^{2} + 33\cdot 61^{3} + 25\cdot 61^{4} +O(61^{5})$
$r_{ 2 }$	$=$	$17 + 57\cdot 61 + 59\cdot 61^{2} + 38\cdot 61^{3} + 34\cdot 61^{4} +O(61^{5})$
$r_{ 3 }$	$=$	$40 + 53\cdot 61 + 24\cdot 61^{3} + 5\cdot 61^{4} +O(61^{5})$
$r_{ 4 }$	$=$	$55 + 4\cdot 61 + 59\cdot 61^{2} + 25\cdot 61^{3} + 56\cdot 61^{4} +O(61^{5})$

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	Complex conjugation
$1$	$1$	$()$	$2$	✓
$1$	$2$	$(1,2)(3,4)$	$-2$
$2$	$2$	$(1,3)(2,4)$	$0$
$2$	$2$	$(1,2)$	$0$
$2$	$4$	$(1,4,2,3)$	$0$