Galois orbit of Artin representations 4.2e17.8t21.3

• Label: 4.2e17.8t21.3
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 2^{17}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$131072= 2^{17} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{6} + 10 x^{4} + 4 x^{2} + 1 $ 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 $\Q_{ 337 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 46 + 67\cdot 337 + 228\cdot 337^{2} + 59\cdot 337^{3} + 135\cdot 337^{4} + 21\cdot 337^{5} + 179\cdot 337^{6} +O\left(337^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 89 + 236\cdot 337 + 82\cdot 337^{2} + 40\cdot 337^{3} + 18\cdot 337^{4} + 51\cdot 337^{5} + 68\cdot 337^{6} +O\left(337^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 93 + 55\cdot 337 + 285\cdot 337^{2} + 10\cdot 337^{3} + 17\cdot 337^{4} + 84\cdot 337^{5} + 333\cdot 337^{6} +O\left(337^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 114 + 110\cdot 337 + 297\cdot 337^{2} + 214\cdot 337^{3} + 81\cdot 337^{4} + 327\cdot 337^{5} + 279\cdot 337^{6} +O\left(337^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 223 + 226\cdot 337 + 39\cdot 337^{2} + 122\cdot 337^{3} + 255\cdot 337^{4} + 9\cdot 337^{5} + 57\cdot 337^{6} +O\left(337^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 244 + 281\cdot 337 + 51\cdot 337^{2} + 326\cdot 337^{3} + 319\cdot 337^{4} + 252\cdot 337^{5} + 3\cdot 337^{6} +O\left(337^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 248 + 100\cdot 337 + 254\cdot 337^{2} + 296\cdot 337^{3} + 318\cdot 337^{4} + 285\cdot 337^{5} + 268\cdot 337^{6} +O\left(337^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 291 + 269\cdot 337 + 108\cdot 337^{2} + 277\cdot 337^{3} + 201\cdot 337^{4} + 315\cdot 337^{5} + 157\cdot 337^{6} +O\left(337^{ 7 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.