Artin representation 2.2e3_7_37e2.4t3.1c1

• Label: 2.2e3_7_37e2.4t3.1c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 2^{3} \cdot 7 \cdot 37^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$76664= 2^{3} \cdot 7 \cdot 37^{2} $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} + 19 x^{2} + 56 x - 622 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.2e3_7.2t1.2c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 12 + 47\cdot 79 + 62\cdot 79^{2} + 79^{3} + 50\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 14 + 63\cdot 79 + 22\cdot 79^{2} + 22\cdot 79^{3} + 53\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 59 + 53\cdot 79 + 47\cdot 79^{2} + 43\cdot 79^{3} + 26\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 75 + 72\cdot 79 + 24\cdot 79^{2} + 11\cdot 79^{3} + 28\cdot 79^{4} +O\left(79^{ 5 }\right)$

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

Cycle notation

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

$(1,3)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 4 }$	Character value
$1$	$1$	$()$	$2$
$1$	$2$	$(1,3)(2,4)$	$-2$
$2$	$2$	$(1,2)(3,4)$	$0$
$2$	$2$	$(2,4)$	$0$
$2$	$4$	$(1,2,3,4)$	$0$

The blue line marks the conjugacy class containing complex conjugation.