Artin representation 2.5_839.4t3.2c1

• Label: 2.5_839.4t3.2c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 5 \cdot 839 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$4195= 5 \cdot 839 $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} + 10 x^{2} - 9 x + 230 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.5_839.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 211 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 97 + 124\cdot 211 + 56\cdot 211^{2} + 150\cdot 211^{3} + 128\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 102 + 181\cdot 211 + 151\cdot 211^{2} + 161\cdot 211^{3} + 144\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 110 + 29\cdot 211 + 59\cdot 211^{2} + 49\cdot 211^{3} + 66\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 115 + 86\cdot 211 + 154\cdot 211^{2} + 60\cdot 211^{3} + 82\cdot 211^{4} +O\left(211^{ 5 }\right)$

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

Cycle notation

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

$(2,3)$

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.