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

• Label: 4.2e6_17e3.8t30.8
• 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} - 2 x^{6} - 4 x^{5} + 4 x^{4} + 4 x^{3} + x^{2} - 6 x - 2 $ 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:

$r_{ 1 }$	$=$	$ 29 + 26\cdot 409 + 69\cdot 409^{2} + 95\cdot 409^{3} + 29\cdot 409^{4} + 274\cdot 409^{5} + 89\cdot 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 32 + 155\cdot 409 + 254\cdot 409^{2} + 300\cdot 409^{3} + 59\cdot 409^{4} + 33\cdot 409^{5} + 207\cdot 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 140 + 106\cdot 409 + 284\cdot 409^{2} + 103\cdot 409^{3} + 397\cdot 409^{4} + 256\cdot 409^{5} + 248\cdot 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 165 + 96\cdot 409 + 23\cdot 409^{2} + 312\cdot 409^{3} + 226\cdot 409^{4} + 123\cdot 409^{5} + 58\cdot 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 183 + 131\cdot 409 + 62\cdot 409^{2} + 110\cdot 409^{3} + 93\cdot 409^{4} + 387\cdot 409^{5} + 53\cdot 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 301 + 381\cdot 409 + 371\cdot 409^{2} + 276\cdot 409^{3} + 285\cdot 409^{4} + 356\cdot 409^{5} + 189\cdot 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 385 + 256\cdot 409 + 84\cdot 409^{2} + 259\cdot 409^{3} + 46\cdot 409^{4} + 381\cdot 409^{5} + 159\cdot 409^{6} +O\left(409^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 401 + 72\cdot 409 + 77\cdot 409^{2} + 178\cdot 409^{3} + 88\cdot 409^{4} + 232\cdot 409^{5} + 219\cdot 409^{6} +O\left(409^{ 7 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.