Galois orbit of Artin representations 4.2e4_17e2_29e2.8t15.2

• Label: 4.2e4_17e2_29e2.8t15.2
• Dimension: 4
• Group: $Z_8 : Z_8^\times$
• Conductor: $ 2^{4} \cdot 17^{2} \cdot 29^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$Z_8 : Z_8^\times$
Conductor:	$3888784= 2^{4} \cdot 17^{2} \cdot 29^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} - 7 x^{6} + 14 x^{5} + 15 x^{4} - 48 x^{3} - 17 x^{2} + 96 x + 73 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$Z_8 : Z_8^\times$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 53 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 2 + 32\cdot 53 + 9\cdot 53^{2} + 19\cdot 53^{3} + 47\cdot 53^{4} + 50\cdot 53^{5} + 10\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 7 + 37\cdot 53 + 19\cdot 53^{2} + 47\cdot 53^{3} + 33\cdot 53^{4} + 43\cdot 53^{5} + 39\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 11 + 50\cdot 53 + 11\cdot 53^{2} + 10\cdot 53^{3} + 51\cdot 53^{4} + 48\cdot 53^{5} + 27\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 19 + 8\cdot 53 + 20\cdot 53^{2} + 31\cdot 53^{3} + 44\cdot 53^{4} + 20\cdot 53^{5} + 21\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 20 + 44\cdot 53 + 39\cdot 53^{2} + 47\cdot 53^{3} + 15\cdot 53^{4} + 7\cdot 53^{5} + 22\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 24 + 11\cdot 53 + 21\cdot 53^{2} + 40\cdot 53^{3} + 51\cdot 53^{4} + 26\cdot 53^{5} + 41\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 27 + 5\cdot 53 + 45\cdot 53^{2} + 20\cdot 53^{3} + 32\cdot 53^{4} + 39\cdot 53^{5} + 42\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 51 + 22\cdot 53 + 44\cdot 53^{2} + 47\cdot 53^{3} + 40\cdot 53^{4} + 26\cdot 53^{5} + 5\cdot 53^{6} +O\left(53^{ 7 }\right)$

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

Cycle notation

$(1,7)(2,3)$

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.