Artin representation 2.5_7e2_331.4t3.2c1

• Label: 2.5_7e2_331.4t3.2c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 5 \cdot 7^{2} \cdot 331 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$81095= 5 \cdot 7^{2} \cdot 331 $
Artin number field:	Splitting field of $f= x^{4} - x^{3} + 19 x^{2} - 299 x + 1355 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.5_331.2t1.1c1

Galois action

Roots of defining polynomial

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

Roots:

$r_{ 1 }$	$=$	$ 8 + 58\cdot 71 + 40\cdot 71^{2} + 37\cdot 71^{3} + 25\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 9 + 15\cdot 71 + 32\cdot 71^{2} + 65\cdot 71^{3} + 47\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 14 + 30\cdot 71 + 24\cdot 71^{2} + 70\cdot 71^{3} + 52\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 41 + 38\cdot 71 + 44\cdot 71^{2} + 39\cdot 71^{3} + 15\cdot 71^{4} +O\left(71^{ 5 }\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.