Galois orbit of Artin representations 4.2e12_5e4.8t16.1

• Label: 4.2e12_5e4.8t16.1
• Dimension: 4
• Group: $(C_8:C_2):C_2$
• Conductor: $ 2^{12} \cdot 5^{4}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(C_8:C_2):C_2$
Conductor:	$2560000= 2^{12} \cdot 5^{4} $
Artin number field:	Splitting field of $f= x^{8} - 5 x^{4} + 5 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(C_8:C_2):C_2$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 101 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 3 + 64\cdot 101 + 44\cdot 101^{2} + 62\cdot 101^{3} + 43\cdot 101^{4} + 10\cdot 101^{5} + 98\cdot 101^{6} + 28\cdot 101^{7} +O\left(101^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 30 + 49\cdot 101 + 45\cdot 101^{2} + 74\cdot 101^{3} + 36\cdot 101^{4} + 52\cdot 101^{5} + 30\cdot 101^{6} + 42\cdot 101^{7} +O\left(101^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 45 + 16\cdot 101 + 83\cdot 101^{2} + 73\cdot 101^{3} + 2\cdot 101^{4} + 15\cdot 101^{5} + 88\cdot 101^{6} + 77\cdot 101^{7} +O\left(101^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 46 + 86\cdot 101 + 97\cdot 101^{2} + 55\cdot 101^{3} + 16\cdot 101^{5} + 22\cdot 101^{6} + 27\cdot 101^{7} +O\left(101^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 55 + 14\cdot 101 + 3\cdot 101^{2} + 45\cdot 101^{3} + 100\cdot 101^{4} + 84\cdot 101^{5} + 78\cdot 101^{6} + 73\cdot 101^{7} +O\left(101^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 56 + 84\cdot 101 + 17\cdot 101^{2} + 27\cdot 101^{3} + 98\cdot 101^{4} + 85\cdot 101^{5} + 12\cdot 101^{6} + 23\cdot 101^{7} +O\left(101^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 71 + 51\cdot 101 + 55\cdot 101^{2} + 26\cdot 101^{3} + 64\cdot 101^{4} + 48\cdot 101^{5} + 70\cdot 101^{6} + 58\cdot 101^{7} +O\left(101^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 98 + 36\cdot 101 + 56\cdot 101^{2} + 38\cdot 101^{3} + 57\cdot 101^{4} + 90\cdot 101^{5} + 2\cdot 101^{6} + 72\cdot 101^{7} +O\left(101^{ 8 }\right)$

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

Cycle notation

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

$(3,6)(4,5)$

$(2,7)(3,6)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.