Artin representation 2.5_311.4t3.2c1

• Label: 2.5_311.4t3.2c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 5 \cdot 311 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$1555= 5 \cdot 311 $
Artin number field:	Splitting field of $f= x^{4} + 3 x^{2} + 80 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.5_311.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 389 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 27 + 168\cdot 389 + 96\cdot 389^{2} + 286\cdot 389^{3} + 371\cdot 389^{4} +O\left(389^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 128 + 323\cdot 389 + 119\cdot 389^{2} + 197\cdot 389^{3} + 32\cdot 389^{4} +O\left(389^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 261 + 65\cdot 389 + 269\cdot 389^{2} + 191\cdot 389^{3} + 356\cdot 389^{4} +O\left(389^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 362 + 220\cdot 389 + 292\cdot 389^{2} + 102\cdot 389^{3} + 17\cdot 389^{4} +O\left(389^{ 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.