Artin representation 2.5015.4t3.a.a

• Label: 2.5015.4t3.a.a
• Dimension: $2$
• Group: $D_{4}$
• Conductor: $5015$
• Root number: $1$
• Indicator: $1$

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$5015$$\medspace = 5 \cdot 17 \cdot 59$
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin stem field:	Galois closure of 4.2.426275.1
Galois orbit size:	$1$
Smallest permutation container:	$D_{4}$
Parity:	odd
Determinant:	1.5015.2t1.a.a
Projective image:	$C_2^2$
Projective field:	Galois closure of $\Q(\sqrt{-59}, \sqrt{85})$

Defining polynomial

$f(x)$

$=$

$x^{4} - x^{3} - 16x^{2} + 40x - 15$ .

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$5 + 4\cdot 19 + 7\cdot 19^{2} + 17\cdot 19^{3} + 19^{4} + 15\cdot 19^{5} +O(19^{6})$
$r_{ 2 }$	$=$	$7 + 6\cdot 19 + 7\cdot 19^{2} + 9\cdot 19^{3} + 12\cdot 19^{4} + 5\cdot 19^{5} +O(19^{6})$
$r_{ 3 }$	$=$	$11 + 18\cdot 19 + 10\cdot 19^{2} + 12\cdot 19^{3} + 18\cdot 19^{4} + 2\cdot 19^{5} +O(19^{6})$
$r_{ 4 }$	$=$	$16 + 8\cdot 19 + 12\cdot 19^{2} + 17\cdot 19^{3} + 4\cdot 19^{4} + 14\cdot 19^{5} +O(19^{6})$

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