Artin representation 2.2e4_5e2.4t3.2c1

• Label: 2.2e4_5e2.4t3.2c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 2^{4} \cdot 5^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$400= 2^{4} \cdot 5^{2} $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} + 4 x^{2} + 2 x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.2e2.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 59 + 54\cdot 101 + 68\cdot 101^{2} + 71\cdot 101^{3} + 95\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 77 + 46\cdot 101 + 82\cdot 101^{2} + 97\cdot 101^{3} + 39\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 80 + 17\cdot 101 + 5\cdot 101^{2} + 87\cdot 101^{3} + 82\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 89 + 82\cdot 101 + 45\cdot 101^{2} + 46\cdot 101^{3} + 84\cdot 101^{4} +O\left(101^{ 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.