Galois orbit of Artin representations 4.13e2_17e3.8t21.1

• Label: 4.13e2_17e3.8t21.1
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 13^{2} \cdot 17^{3}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$830297= 13^{2} \cdot 17^{3} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + 5 x^{6} - 2 x^{5} - 9 x^{4} + 2 x^{3} + 5 x^{2} + 2 x + 1 $ 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 $\Q_{ 101 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 6 + 93\cdot 101 + 20\cdot 101^{2} + 88\cdot 101^{3} + 54\cdot 101^{4} + 22\cdot 101^{5} +O\left(101^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 27 + 64\cdot 101 + 94\cdot 101^{2} + 65\cdot 101^{3} + 7\cdot 101^{4} + 99\cdot 101^{5} +O\left(101^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 38 + 19\cdot 101 + 82\cdot 101^{2} + 57\cdot 101^{3} + 65\cdot 101^{4} + 52\cdot 101^{5} +O\left(101^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 76 + 44\cdot 101 + 71\cdot 101^{2} + 29\cdot 101^{3} + 40\cdot 101^{4} + 87\cdot 101^{5} +O\left(101^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 84 + 27\cdot 101 + 29\cdot 101^{2} + 56\cdot 101^{3} + 3\cdot 101^{4} + 15\cdot 101^{5} +O\left(101^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 86 + 16\cdot 101 + 57\cdot 101^{2} + 92\cdot 101^{3} + 34\cdot 101^{4} + 65\cdot 101^{5} +O\left(101^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 93 + 27\cdot 101 + 11\cdot 101^{2} + 47\cdot 101^{3} + 59\cdot 101^{4} + 3\cdot 101^{5} +O\left(101^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 97 + 8\cdot 101 + 37\cdot 101^{2} + 67\cdot 101^{3} + 36\cdot 101^{4} + 58\cdot 101^{5} +O\left(101^{ 6 }\right)$

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

Cycle notation

$(1,5)(2,6)$

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.