Artin representation 2.3_13_37.4t3.4c1

• Label: 2.3_13_37.4t3.4c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 3 \cdot 13 \cdot 37 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$1443= 3 \cdot 13 \cdot 37 $
Artin number field:	Splitting field of $f= x^{4} - x^{3} - 10 x^{2} + 7 x + 31 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.3_13_37.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 22\cdot 31 + 22\cdot 31^{2} + 16\cdot 31^{3} + 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 15 + 4\cdot 31 + 23\cdot 31^{2} + 2\cdot 31^{3} + 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 22 + 9\cdot 31 + 31^{2} + 6\cdot 31^{3} + 17\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 26 + 25\cdot 31 + 14\cdot 31^{2} + 5\cdot 31^{3} + 11\cdot 31^{4} +O\left(31^{ 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.