Artin representation 4.2e4_3e2_7e2_29e2.8t22.5c1

• Label: 4.2e4_3e2_7e2_29e2.8t22.5c1
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 29^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$5934096= 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 29^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} - 9 x^{6} + 24 x^{5} + 36 x^{4} - 110 x^{3} - 109 x^{2} + 184 x + 193 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3 : D_4 $
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 8 + 2\cdot 109 + 90\cdot 109^{2} + 34\cdot 109^{3} + 36\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 16 + 37\cdot 109 + 7\cdot 109^{2} + 49\cdot 109^{3} + 78\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 23 + 55\cdot 109 + 50\cdot 109^{2} + 60\cdot 109^{3} + 12\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 32 + 55\cdot 109 + 91\cdot 109^{2} + 94\cdot 109^{3} + 93\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 37 + 53\cdot 109 + 99\cdot 109^{2} + 64\cdot 109^{3} + 61\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 63 + 14\cdot 109 + 70\cdot 109^{2} + 73\cdot 109^{3} + 90\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 65 + 5\cdot 109 + 20\cdot 109^{2} + 47\cdot 109^{3} + 27\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 85 + 103\cdot 109 + 6\cdot 109^{2} + 11\cdot 109^{3} + 35\cdot 109^{4} +O\left(109^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation

$(1,4,3,7)(2,5,6,8)$

$(1,4)(2,8)(3,7)(5,6)$

$(1,4,3,7)(2,8,6,5)$

$(1,7)(2,8)(3,4)(5,6)$

$(1,5)(2,4)(3,8)(6,7)$

$(1,5)(2,7)(3,8)(4,6)$

$(1,3)(4,7)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character value
$1$	$1$	$()$	$4$
$1$	$2$	$(1,3)(2,6)(4,7)(5,8)$	$-4$
$2$	$2$	$(1,4)(2,8)(3,7)(5,6)$	$0$
$2$	$2$	$(1,5)(2,4)(3,8)(6,7)$	$0$
$2$	$2$	$(1,5)(2,7)(3,8)(4,6)$	$0$
$2$	$2$	$(1,2)(3,6)(4,8)(5,7)$	$0$
$2$	$2$	$(1,3)(4,7)$	$0$
$2$	$2$	$(1,7)(2,8)(3,4)(5,6)$	$0$
$2$	$2$	$(1,6)(2,3)(4,8)(5,7)$	$0$
$2$	$2$	$(4,7)(5,8)$	$0$
$2$	$2$	$(1,3)(5,8)$	$0$
$2$	$4$	$(1,4,3,7)(2,5,6,8)$	$0$
$2$	$4$	$(1,2,3,6)(4,5,7,8)$	$0$
$2$	$4$	$(1,7,3,4)(2,5,6,8)$	$0$
$2$	$4$	$(1,8,3,5)(2,4,6,7)$	$0$
$2$	$4$	$(1,8,3,5)(2,7,6,4)$	$0$
$2$	$4$	$(1,6,3,2)(4,5,7,8)$	$0$

The blue line marks the conjugacy class containing complex conjugation.