Artin representation 4.2e4_17e2_29e2.8t15.1c1

• Label: 4.2e4_17e2_29e2.8t15.1c1
• Dimension: 4
• Group: $Z_8 : Z_8^\times$
• Conductor: $ 2^{4} \cdot 17^{2} \cdot 29^{2}$
• Root number: 1
• 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} - 3 x^{6} + 10 x^{5} - 6 x^{4} - 28 x^{3} + 46 x^{2} + 110 x + 53 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$Z_8 : Z_8^\times$
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

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

Roots:

$r_{ 1 }$	$=$	$ 13\cdot 53 + 24\cdot 53^{2} + 5\cdot 53^{3} + 18\cdot 53^{4} + 44\cdot 53^{5} + 13\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 2 + 4\cdot 53 + 43\cdot 53^{2} + 18\cdot 53^{3} + 17\cdot 53^{4} + 6\cdot 53^{5} + 41\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 29 + 24\cdot 53 + 30\cdot 53^{2} + 34\cdot 53^{3} + 8\cdot 53^{4} + 46\cdot 53^{5} + 39\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 30 + 34\cdot 53 + 25\cdot 53^{2} + 43\cdot 53^{3} + 19\cdot 53^{4} + 41\cdot 53^{5} + 39\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 34 + 24\cdot 53 + 38\cdot 53^{2} + 12\cdot 53^{3} + 37\cdot 53^{4} + 30\cdot 53^{5} + 51\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 36 + 6\cdot 53 + 26\cdot 53^{2} + 13\cdot 53^{3} + 48\cdot 53^{4} + 16\cdot 53^{5} + 28\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 41 + 52\cdot 53 + 5\cdot 53^{2} + 14\cdot 53^{3} + 12\cdot 53^{4} + 51\cdot 53^{5} + 27\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 42 + 51\cdot 53 + 17\cdot 53^{2} + 16\cdot 53^{3} + 50\cdot 53^{4} + 27\cdot 53^{5} + 22\cdot 53^{6} +O\left(53^{ 7 }\right)$

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

Cycle notation

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

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

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

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

$(1,3)(4,7)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.