Galois orbit of Artin representations 4.2e14_5e3.8t21.1

• Label: 4.2e14_5e3.8t21.1
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 2^{14} \cdot 5^{3}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$2048000= 2^{14} \cdot 5^{3} $
Artin number field:	Splitting field of $f= x^{8} - 8 x^{6} + 30 x^{4} - 40 x^{2} + 25 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3: C_4$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 109 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 5 + 15\cdot 109 + 49\cdot 109^{2} + 9\cdot 109^{3} + 84\cdot 109^{4} + 4\cdot 109^{5} + 48\cdot 109^{6} + 56\cdot 109^{7} +O\left(109^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 15 + 89\cdot 109 + 46\cdot 109^{2} + 40\cdot 109^{3} + 10\cdot 109^{4} + 74\cdot 109^{5} + 60\cdot 109^{6} + 60\cdot 109^{7} +O\left(109^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 26 + 64\cdot 109 + 29\cdot 109^{2} + 58\cdot 109^{3} + 83\cdot 109^{4} + 90\cdot 109^{5} + 72\cdot 109^{6} + 12\cdot 109^{7} +O\left(109^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 45 + 107\cdot 109 + 44\cdot 109^{2} + 109^{3} + 64\cdot 109^{4} + 101\cdot 109^{5} + 10\cdot 109^{6} + 95\cdot 109^{7} +O\left(109^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 64 + 109 + 64\cdot 109^{2} + 107\cdot 109^{3} + 44\cdot 109^{4} + 7\cdot 109^{5} + 98\cdot 109^{6} + 13\cdot 109^{7} +O\left(109^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 83 + 44\cdot 109 + 79\cdot 109^{2} + 50\cdot 109^{3} + 25\cdot 109^{4} + 18\cdot 109^{5} + 36\cdot 109^{6} + 96\cdot 109^{7} +O\left(109^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 94 + 19\cdot 109 + 62\cdot 109^{2} + 68\cdot 109^{3} + 98\cdot 109^{4} + 34\cdot 109^{5} + 48\cdot 109^{6} + 48\cdot 109^{7} +O\left(109^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 104 + 93\cdot 109 + 59\cdot 109^{2} + 99\cdot 109^{3} + 24\cdot 109^{4} + 104\cdot 109^{5} + 60\cdot 109^{6} + 52\cdot 109^{7} +O\left(109^{ 8 }\right)$

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

Cycle notation

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

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

$(2,7)(4,5)$

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

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$
$1$	$1$	$()$	$4$
$1$	$2$	$(1,8)(2,7)(3,6)(4,5)$	$-4$
$2$	$2$	$(1,6)(2,5)(3,8)(4,7)$	$0$
$2$	$2$	$(1,6)(2,4)(3,8)(5,7)$	$0$
$2$	$2$	$(2,7)(4,5)$	$0$
$4$	$2$	$(1,7)(2,8)(3,5)(4,6)$	$0$
$4$	$4$	$(1,4,8,5)(2,6,7,3)$	$0$
$4$	$4$	$(1,2,6,4)(3,5,8,7)$	$0$
$4$	$4$	$(1,4,6,2)(3,7,8,5)$	$0$
$4$	$4$	$(1,8)(2,5,7,4)$	$0$
$4$	$4$	$(1,8)(2,4,7,5)$	$0$

The blue line marks the conjugacy class containing complex conjugation.