Artin representation 2.2e3_7e2_11.4t3.2c1

• Label: 2.2e3_7e2_11.4t3.2c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 2^{3} \cdot 7^{2} \cdot 11 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$4312= 2^{3} \cdot 7^{2} \cdot 11 $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} - 23 x^{2} + 24 x - 10 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.2e3_11.2t1.2c1

Galois action

Roots of defining polynomial

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

Roots:

$r_{ 1 }$	$=$	$ 15 + 70\cdot 71 + 23\cdot 71^{3} + 37\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 16 + 58\cdot 71 + 34\cdot 71^{2} + 60\cdot 71^{3} + 25\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 56 + 12\cdot 71 + 36\cdot 71^{2} + 10\cdot 71^{3} + 45\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 57 + 70\cdot 71^{2} + 47\cdot 71^{3} + 33\cdot 71^{4} +O\left(71^{ 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.