Galois orbit of Artin representations 4.2e17_3e2.8t21.4

• Label: 4.2e17_3e2.8t21.4
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 2^{17} \cdot 3^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$1179648= 2^{17} \cdot 3^{2} $
Artin number field:	Splitting field of $f= x^{8} - 8 x^{6} - 8 x^{5} + 24 x^{4} + 64 x^{3} + 56 x^{2} + 16 x + 2 $ 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 $\Q_{ 89 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 4 + 73\cdot 89 + 16\cdot 89^{3} + 18\cdot 89^{4} + 13\cdot 89^{6} + 45\cdot 89^{7} +O\left(89^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 13 + 73\cdot 89 + 56\cdot 89^{2} + 5\cdot 89^{3} + 28\cdot 89^{4} + 16\cdot 89^{5} + 80\cdot 89^{6} + 14\cdot 89^{7} +O\left(89^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 14 + 9\cdot 89 + 49\cdot 89^{2} + 52\cdot 89^{3} + 67\cdot 89^{4} + 71\cdot 89^{5} + 48\cdot 89^{6} + 38\cdot 89^{7} +O\left(89^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 27 + 70\cdot 89 + 33\cdot 89^{2} + 41\cdot 89^{3} + 42\cdot 89^{4} + 88\cdot 89^{5} + 89^{6} + 89^{7} +O\left(89^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 63 + 32\cdot 89 + 52\cdot 89^{2} + 54\cdot 89^{3} + 14\cdot 89^{4} + 61\cdot 89^{5} + 36\cdot 89^{6} + 68\cdot 89^{7} +O\left(89^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 75 + 89 + 35\cdot 89^{2} + 76\cdot 89^{3} + 3\cdot 89^{4} + 12\cdot 89^{5} + 59\cdot 89^{6} + 4\cdot 89^{7} +O\left(89^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 78 + 4\cdot 89 + 2\cdot 89^{2} + 49\cdot 89^{3} + 77\cdot 89^{4} + 10\cdot 89^{5} + 84\cdot 89^{6} + 22\cdot 89^{7} +O\left(89^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 82 + 89 + 37\cdot 89^{2} + 60\cdot 89^{3} + 14\cdot 89^{4} + 6\cdot 89^{5} + 32\cdot 89^{6} + 71\cdot 89^{7} +O\left(89^{ 8 }\right)$

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

Cycle notation

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

$(2,5)(4,6)$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.