Artin representation 2.2e3_7e2.4t3.1c1

• Label: 2.2e3_7e2.4t3.1c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 2^{3} \cdot 7^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$392= 2^{3} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} + 5 x^{2} - 4 x - 10 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.2e3.2t1.2c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 8 + 25\cdot 67 + 56\cdot 67^{2} + 45\cdot 67^{3} + 36\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 19 + 6\cdot 67 + 28\cdot 67^{2} + 67^{3} + 25\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 49 + 60\cdot 67 + 38\cdot 67^{2} + 65\cdot 67^{3} + 41\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 60 + 41\cdot 67 + 10\cdot 67^{2} + 21\cdot 67^{3} + 30\cdot 67^{4} +O\left(67^{ 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.