Galois orbit of Artin representations 4.2e13_17e2.8t21.4

• Label: 4.2e13_17e2.8t21.4
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 2^{13} \cdot 17^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$2367488= 2^{13} \cdot 17^{2} $
Artin number field:	Splitting field of $f= x^{8} - 8 x^{6} + 34 x^{4} - 72 x^{2} + 64 $ 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 $\Q_{ 359 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 51 + 274\cdot 359 + 254\cdot 359^{2} + 172\cdot 359^{3} + 309\cdot 359^{4} + 328\cdot 359^{5} + 308\cdot 359^{6} +O\left(359^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 73 + 346\cdot 359 + 104\cdot 359^{2} + 170\cdot 359^{3} + 150\cdot 359^{4} + 93\cdot 359^{5} + 122\cdot 359^{6} +O\left(359^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 135 + 234\cdot 359 + 228\cdot 359^{2} + 357\cdot 359^{3} + 97\cdot 359^{4} + 333\cdot 359^{5} + 3\cdot 359^{6} +O\left(359^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 142 + 349\cdot 359 + 68\cdot 359^{2} + 220\cdot 359^{3} + 163\cdot 359^{4} + 98\cdot 359^{5} + 81\cdot 359^{6} +O\left(359^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 217 + 9\cdot 359 + 290\cdot 359^{2} + 138\cdot 359^{3} + 195\cdot 359^{4} + 260\cdot 359^{5} + 277\cdot 359^{6} +O\left(359^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 224 + 124\cdot 359 + 130\cdot 359^{2} + 359^{3} + 261\cdot 359^{4} + 25\cdot 359^{5} + 355\cdot 359^{6} +O\left(359^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 286 + 12\cdot 359 + 254\cdot 359^{2} + 188\cdot 359^{3} + 208\cdot 359^{4} + 265\cdot 359^{5} + 236\cdot 359^{6} +O\left(359^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 308 + 84\cdot 359 + 104\cdot 359^{2} + 186\cdot 359^{3} + 49\cdot 359^{4} + 30\cdot 359^{5} + 50\cdot 359^{6} +O\left(359^{ 7 }\right)$

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

Cycle notation

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

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

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

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

$(1,8)(3,6)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.