Galois orbit of Artin representations 4.2e4_13e2_17e3.8t21.6

• Label: 4.2e4_13e2_17e3.8t21.6
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 2^{4} \cdot 13^{2} \cdot 17^{3}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$13284752= 2^{4} \cdot 13^{2} \cdot 17^{3} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} - x^{6} - 34 x^{3} + 11 x^{2} + 188 x + 157 $ 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 $\Q_{ 157 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 81\cdot 157 + 120\cdot 157^{2} + 138\cdot 157^{3} + 118\cdot 157^{4} + 98\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 9 + 136\cdot 157 + 153\cdot 157^{2} + 20\cdot 157^{3} + 49\cdot 157^{4} + 97\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 12 + 51\cdot 157 + 14\cdot 157^{2} + 96\cdot 157^{3} + 36\cdot 157^{4} + 79\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 19 + 133\cdot 157 + 148\cdot 157^{2} + 133\cdot 157^{3} + 80\cdot 157^{4} + 6\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 37 + 69\cdot 157 + 28\cdot 157^{2} + 24\cdot 157^{3} + 115\cdot 157^{4} + 112\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 124 + 92\cdot 157 + 32\cdot 157^{2} + 35\cdot 157^{3} + 56\cdot 157^{4} + 33\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 127 + 48\cdot 157 + 30\cdot 157^{2} + 102\cdot 157^{3} + 77\cdot 157^{4} + 129\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 145 + 15\cdot 157 + 99\cdot 157^{2} + 76\cdot 157^{3} + 93\cdot 157^{4} + 70\cdot 157^{5} +O\left(157^{ 6 }\right)$

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

Cycle notation

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

$(2,6)(5,8)$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.