Galois orbit of Artin representations 4.2e12_3e2_7e2.8t15.4

• Label: 4.2e12_3e2_7e2.8t15.4
• Dimension: 4
• Group: $Z_8 : Z_8^\times$
• Conductor: $ 2^{12} \cdot 3^{2} \cdot 7^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$Z_8 : Z_8^\times$
Conductor:	$1806336= 2^{12} \cdot 3^{2} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{8} + 2 x^{6} - x^{4} - 8 x^{2} + 7 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$Z_8 : Z_8^\times$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 337 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 32 + 15\cdot 337 + 150\cdot 337^{2} + 292\cdot 337^{3} + 16\cdot 337^{4} + 272\cdot 337^{5} + 294\cdot 337^{6} +O\left(337^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 49 + 168\cdot 337 + 156\cdot 337^{2} + 233\cdot 337^{3} + 30\cdot 337^{4} + 96\cdot 337^{5} + 290\cdot 337^{6} +O\left(337^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 68 + 304\cdot 337 + 38\cdot 337^{2} + 294\cdot 337^{3} + 21\cdot 337^{4} + 68\cdot 337^{5} + 256\cdot 337^{6} +O\left(337^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 126 + 268\cdot 337 + 147\cdot 337^{2} + 180\cdot 337^{3} + 72\cdot 337^{4} + 237\cdot 337^{5} + 253\cdot 337^{6} +O\left(337^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 211 + 68\cdot 337 + 189\cdot 337^{2} + 156\cdot 337^{3} + 264\cdot 337^{4} + 99\cdot 337^{5} + 83\cdot 337^{6} +O\left(337^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 269 + 32\cdot 337 + 298\cdot 337^{2} + 42\cdot 337^{3} + 315\cdot 337^{4} + 268\cdot 337^{5} + 80\cdot 337^{6} +O\left(337^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 288 + 168\cdot 337 + 180\cdot 337^{2} + 103\cdot 337^{3} + 306\cdot 337^{4} + 240\cdot 337^{5} + 46\cdot 337^{6} +O\left(337^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 305 + 321\cdot 337 + 186\cdot 337^{2} + 44\cdot 337^{3} + 320\cdot 337^{4} + 64\cdot 337^{5} + 42\cdot 337^{6} +O\left(337^{ 7 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.