Artin representation 2.5_101.4t3.2c1

• Label: 2.5_101.4t3.2c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 5 \cdot 101 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$505= 5 \cdot 101 $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 5 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Even
Determinant:	1.5_101.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 21 + 46\cdot 71 + 34\cdot 71^{2} + 47\cdot 71^{3} + 54\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 26 + 56\cdot 71 + 41\cdot 71^{2} + 59\cdot 71^{3} + 52\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 46 + 14\cdot 71 + 29\cdot 71^{2} + 11\cdot 71^{3} + 18\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 51 + 24\cdot 71 + 36\cdot 71^{2} + 23\cdot 71^{3} + 16\cdot 71^{4} +O\left(71^{ 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.