Artin representation 2.3_7_127.4t3.1c1

• Label: 2.3_7_127.4t3.1c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 3 \cdot 7 \cdot 127 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$2667= 3 \cdot 7 \cdot 127 $
Artin number field:	Splitting field of $f= x^{4} + 11 x^{2} - 192 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.3_7_127.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 223 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 63 + 34\cdot 223 + 103\cdot 223^{2} + 201\cdot 223^{3} + 64\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 82 + 115\cdot 223 + 182\cdot 223^{2} + 75\cdot 223^{3} + 97\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 141 + 107\cdot 223 + 40\cdot 223^{2} + 147\cdot 223^{3} + 125\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 160 + 188\cdot 223 + 119\cdot 223^{2} + 21\cdot 223^{3} + 158\cdot 223^{4} +O\left(223^{ 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.