Galois orbit of Artin representations 4.5e2_11_29e3.8t35.5

• Label: 4.5e2_11_29e3.8t35.5
• Dimension: 4
• Group: $C_2 \wr C_2\wr C_2$
• Conductor: $ 5^{2} \cdot 11 \cdot 29^{3}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2 \wr C_2\wr C_2$
Conductor:	$6706975= 5^{2} \cdot 11 \cdot 29^{3} $
Artin number field:	Splitting field of $f= x^{8} - x^{7} + x^{6} - 2 x^{5} + x^{4} - 2 x^{3} + x^{2} - x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2 \wr C_2\wr C_2$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 929 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 188 + 797\cdot 929 + 563\cdot 929^{2} + 216\cdot 929^{3} + 272\cdot 929^{4} + 544\cdot 929^{5} +O\left(929^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 193 + 272\cdot 929 + 459\cdot 929^{2} + 885\cdot 929^{3} + 861\cdot 929^{4} + 111\cdot 929^{5} +O\left(929^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 284 + 798\cdot 929 + 254\cdot 929^{2} + 690\cdot 929^{3} + 92\cdot 929^{4} + 333\cdot 929^{5} +O\left(929^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 344 + 590\cdot 929 + 14\cdot 929^{2} + 376\cdot 929^{3} + 419\cdot 929^{4} + 237\cdot 929^{5} +O\left(929^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 451 + 633\cdot 929 + 928\cdot 929^{2} + 823\cdot 929^{3} + 466\cdot 929^{4} + 746\cdot 929^{5} +O\left(929^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 552 + 148\cdot 929 + 32\cdot 929^{2} + 371\cdot 929^{3} + 702\cdot 929^{4} + 225\cdot 929^{5} +O\left(929^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 845 + 34\cdot 929 + 232\cdot 929^{2} + 446\cdot 929^{3} + 49\cdot 929^{4} + 218\cdot 929^{5} +O\left(929^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 860 + 440\cdot 929 + 301\cdot 929^{2} + 835\cdot 929^{3} + 850\cdot 929^{4} + 369\cdot 929^{5} +O\left(929^{ 6 }\right)$

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

Cycle notation

$(2,6)(3,8)$

$(1,7)$

$(4,5)$

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

$(6,8)$

$(2,3)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$
$1$	$1$	$()$	$4$
$1$	$2$	$(1,7)(2,3)(4,5)(6,8)$	$-4$
$2$	$2$	$(1,7)(4,5)$	$0$
$4$	$2$	$(1,7)$	$2$
$4$	$2$	$(1,7)(6,8)$	$0$
$4$	$2$	$(1,4)(2,6)(3,8)(5,7)$	$0$
$4$	$2$	$(1,4)(5,7)$	$-2$
$4$	$2$	$(1,7)(2,6)(3,8)(4,5)$	$2$
$4$	$2$	$(1,7)(2,3)(4,5)$	$-2$
$8$	$2$	$(1,6)(2,4)(3,5)(7,8)$	$0$
$8$	$2$	$(1,7)(2,6)(3,8)$	$0$
$4$	$4$	$(1,5,7,4)(2,6,3,8)$	$0$
$4$	$4$	$(1,5,7,4)$	$-2$
$4$	$4$	$(1,7)(2,8,3,6)(4,5)$	$2$
$8$	$4$	$(1,8,7,6)(2,4,3,5)$	$0$
$8$	$4$	$(1,5,7,4)(2,6)(3,8)$	$0$
$8$	$4$	$(1,5,7,4)(2,3)$	$0$
$16$	$4$	$(1,8,7,6)(2,4)(3,5)$	$0$
$16$	$4$	$(1,2,4,6)(3,5,8,7)$	$0$
$16$	$8$	$(1,3,5,8,7,2,4,6)$	$0$

The blue line marks the conjugacy class containing complex conjugation.