Galois orbit of Artin representations 4.2e6_17e3.8t30.6

• Label: 4.2e6_17e3.8t30.6
• Dimension: 4
• Group: $(((C_4 \times C_2): C_2):C_2):C_2$
• Conductor: $ 2^{6} \cdot 17^{3}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(((C_4 \times C_2): C_2):C_2):C_2$
Conductor:	$314432= 2^{6} \cdot 17^{3} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{7} + 9 x^{6} - 12 x^{5} + 7 x^{4} + 10 x^{3} - 22 x^{2} + 16 x - 4 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(((C_4 \times C_2): C_2):C_2):C_2$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 409 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 31 + 94\cdot 409 + 402\cdot 409^{2} + 373\cdot 409^{3} + 167\cdot 409^{4} + 50\cdot 409^{5} + 47\cdot 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 137 + 75\cdot 409 + 37\cdot 409^{2} + 281\cdot 409^{3} + 125\cdot 409^{4} + 292\cdot 409^{5} + 181\cdot 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 151 + 334\cdot 409 + 9\cdot 409^{2} + 45\cdot 409^{3} + 311\cdot 409^{4} + 305\cdot 409^{5} + 255\cdot 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 309 + 127\cdot 409 + 148\cdot 409^{2} + 206\cdot 409^{3} + 311\cdot 409^{4} + 301\cdot 409^{5} + 8\cdot 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 329 + 261\cdot 409 + 257\cdot 409^{2} + 192\cdot 409^{3} + 27\cdot 409^{4} + 160\cdot 409^{5} + 97\cdot 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 335 + 123\cdot 409 + 337\cdot 409^{2} + 291\cdot 409^{3} + 34\cdot 409^{4} + 289\cdot 409^{5} + 62\cdot 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 350 + 69\cdot 409 + 222\cdot 409^{2} + 197\cdot 409^{3} + 194\cdot 409^{4} + 257\cdot 409^{5} + 188\cdot 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 407 + 139\cdot 409 + 221\cdot 409^{2} + 47\cdot 409^{3} + 54\cdot 409^{4} + 388\cdot 409^{5} + 384\cdot 409^{6} +O\left(409^{ 7 }\right)$

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

Cycle notation

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

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

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

$(1,3)(6,7)$

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

$(1,3)(4,5)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.