Galois orbit of Artin representations 4.3e2_13_109.8t35.2

• Label: 4.3e2_13_109.8t35.2
• Dimension: 4
• Group: $C_2 \wr C_2\wr C_2$
• Conductor: $ 3^{2} \cdot 13 \cdot 109 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2 \wr C_2\wr C_2$
Conductor:	$12753= 3^{2} \cdot 13 \cdot 109 $
Artin number field:	Splitting field of $f= x^{8} - x^{7} + x^{6} - 3 x^{5} + x^{4} - 2 x^{3} + 3 x^{2} + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2 \wr C_2\wr C_2$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 823 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 39 + 731\cdot 823 + 15\cdot 823^{2} + 286\cdot 823^{3} + 156\cdot 823^{4} + 761\cdot 823^{5} +O\left(823^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 126 + 533\cdot 823 + 352\cdot 823^{2} + 551\cdot 823^{3} + 586\cdot 823^{4} + 88\cdot 823^{5} +O\left(823^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 157 + 531\cdot 823 + 205\cdot 823^{2} + 500\cdot 823^{3} + 474\cdot 823^{4} + 570\cdot 823^{5} +O\left(823^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 227 + 761\cdot 823 + 375\cdot 823^{2} + 408\cdot 823^{3} + 817\cdot 823^{4} + 122\cdot 823^{5} +O\left(823^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 338 + 423\cdot 823 + 58\cdot 823^{2} + 525\cdot 823^{3} + 692\cdot 823^{4} + 351\cdot 823^{5} +O\left(823^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 413 + 815\cdot 823 + 82\cdot 823^{2} + 389\cdot 823^{3} + 9\cdot 823^{4} + 626\cdot 823^{5} +O\left(823^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 464 + 698\cdot 823 + 420\cdot 823^{2} + 810\cdot 823^{3} + 362\cdot 823^{4} + 746\cdot 823^{5} +O\left(823^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 706 + 443\cdot 823 + 133\cdot 823^{2} + 644\cdot 823^{3} + 191\cdot 823^{4} + 24\cdot 823^{5} +O\left(823^{ 6 }\right)$

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

Cycle notation

$(1,3)$

$(2,8)$

$(5,7)$

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

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

$(4,6)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.