Artin representation 2.2e4_5e2_19e2.4t3.3c1

• Label: 2.2e4_5e2_19e2.4t3.3c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 2^{4} \cdot 5^{2} \cdot 19^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$144400= 2^{4} \cdot 5^{2} \cdot 19^{2} $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} + 4 x^{2} + 92 x + 406 $ 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 $\Q_{ 29 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 3\cdot 29 + 14\cdot 29^{2} + 4\cdot 29^{3} + 18\cdot 29^{4} + 8\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 17 + 23\cdot 29 + 12\cdot 29^{2} + 11\cdot 29^{3} + 26\cdot 29^{4} + 28\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 21 + 28\cdot 29 + 26\cdot 29^{2} + 21\cdot 29^{3} + 15\cdot 29^{4} +O\left(29^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 22 + 2\cdot 29 + 4\cdot 29^{2} + 20\cdot 29^{3} + 26\cdot 29^{4} + 19\cdot 29^{5} +O\left(29^{ 6 }\right)$

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

Cycle notation

$(1,2)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.