Galois orbit of Artin representations 4.2e4_17e2_37e2.8t15.2

• Label: 4.2e4_17e2_37e2.8t15.2
• Dimension: 4
• Group: $Z_8 : Z_8^\times$
• Conductor: $ 2^{4} \cdot 17^{2} \cdot 37^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$Z_8 : Z_8^\times$
Conductor:	$6330256= 2^{4} \cdot 17^{2} \cdot 37^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{7} + 3 x^{6} + 9 x^{4} + 28 x^{3} - 89 x^{2} - 36 x + 149 $ 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:

$r_{ 1 }$	$=$	$ 6 + 29\cdot 53 + 32\cdot 53^{2} + 9\cdot 53^{4} + 43\cdot 53^{5} + 29\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 9 + 44\cdot 53 + 16\cdot 53^{2} + 31\cdot 53^{3} + 41\cdot 53^{4} + 50\cdot 53^{5} + 21\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 13 + 41\cdot 53 + 39\cdot 53^{2} + 27\cdot 53^{3} + 30\cdot 53^{4} + 52\cdot 53^{5} + 33\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 21 + 5\cdot 53 + 3\cdot 53^{2} + 29\cdot 53^{3} + 38\cdot 53^{4} + 22\cdot 53^{5} + 32\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 22 + 39\cdot 53 + 5\cdot 53^{2} + 17\cdot 53^{3} + 24\cdot 53^{4} + 53^{5} + 3\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 24 + 45\cdot 53 + 5\cdot 53^{2} + 52\cdot 53^{3} + 22\cdot 53^{4} + 51\cdot 53^{5} + 11\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 27 + 35\cdot 53 + 53^{2} + 20\cdot 53^{3} + 38\cdot 53^{4} + 13\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 41 + 24\cdot 53 + 34\cdot 53^{3} + 6\cdot 53^{4} + 42\cdot 53^{5} + 12\cdot 53^{6} +O\left(53^{ 7 }\right)$

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

Cycle notation

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

$(1,7)(4,6)$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.