Artin representation 4.2e4_17e2_37e2.8t15.1c1

• Label: 4.2e4_17e2_37e2.8t15.1c1
• Dimension: 4
• Group: $Z_8 : Z_8^\times$
• Conductor: $ 2^{4} \cdot 17^{2} \cdot 37^{2}$
• Root number: 1
• 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} - 2 x^{7} - 5 x^{6} + 24 x^{5} + 16 x^{4} - 72 x^{3} - 36 x^{2} + 74 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:

$r_{ 1 }$	$=$	$ 5\cdot 53 + 40\cdot 53^{2} + 7\cdot 53^{3} + 36\cdot 53^{4} + 9\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 1 + 49\cdot 53 + 34\cdot 53^{2} + 37\cdot 53^{3} + 42\cdot 53^{4} + 35\cdot 53^{5} + 33\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 9 + 53 + 10\cdot 53^{2} + 33\cdot 53^{3} + 48\cdot 53^{4} + 35\cdot 53^{5} + 8\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 18 + 46\cdot 53 + 7\cdot 53^{2} + 22\cdot 53^{3} + 43\cdot 53^{4} + 17\cdot 53^{5} + 45\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 19 + 28\cdot 53 + 33\cdot 53^{2} + 4\cdot 53^{3} + 30\cdot 53^{4} + 40\cdot 53^{5} + 33\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 34 + 51\cdot 53 + 8\cdot 53^{2} + 37\cdot 53^{3} + 37\cdot 53^{4} + 24\cdot 53^{5} + 3\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 36 + 13\cdot 53 + 34\cdot 53^{2} + 40\cdot 53^{3} + 37\cdot 53^{4} + 12\cdot 53^{5} + 29\cdot 53^{6} +O\left(53^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 44 + 16\cdot 53 + 42\cdot 53^{2} + 28\cdot 53^{3} + 41\cdot 53^{4} + 43\cdot 53^{5} + 48\cdot 53^{6} +O\left(53^{ 7 }\right)$

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

Cycle notation

$(1,7)(6,8)$

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.