Galois orbit of Artin representations 4.2e4_3e2_73e3.8t29.2

• Label: 4.2e4_3e2_73e3.8t29.2
• Dimension: 4
• Group: $(((C_4 \times C_2): C_2):C_2):C_2$
• Conductor: $ 2^{4} \cdot 3^{2} \cdot 73^{3}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(((C_4 \times C_2): C_2):C_2):C_2$
Conductor:	$56018448= 2^{4} \cdot 3^{2} \cdot 73^{3} $
Artin number field:	Splitting field of $f= x^{8} - x^{7} + x^{6} + x^{5} - x^{3} + x^{2} - 2 x + 1 $ 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 $\Q_{ 733 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 178 + 382\cdot 733 + 221\cdot 733^{2} + 221\cdot 733^{3} + 178\cdot 733^{4} + 294\cdot 733^{5} +O\left(733^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 238 + 386\cdot 733 + 410\cdot 733^{2} + 670\cdot 733^{3} + 16\cdot 733^{4} + 182\cdot 733^{5} +O\left(733^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 430 + 720\cdot 733 + 291\cdot 733^{2} + 696\cdot 733^{3} + 489\cdot 733^{4} + 50\cdot 733^{5} +O\left(733^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 440 + 74\cdot 733 + 361\cdot 733^{2} + 197\cdot 733^{3} + 367\cdot 733^{4} + 640\cdot 733^{5} +O\left(733^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 534 + 9\cdot 733 + 644\cdot 733^{2} + 364\cdot 733^{3} + 576\cdot 733^{4} + 702\cdot 733^{5} +O\left(733^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 550 + 419\cdot 733 + 727\cdot 733^{2} + 597\cdot 733^{3} + 88\cdot 733^{4} + 326\cdot 733^{5} +O\left(733^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 632 + 410\cdot 733 + 367\cdot 733^{2} + 315\cdot 733^{3} + 671\cdot 733^{4} + 485\cdot 733^{5} +O\left(733^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 664 + 527\cdot 733 + 640\cdot 733^{2} + 600\cdot 733^{3} + 542\cdot 733^{4} + 249\cdot 733^{5} +O\left(733^{ 6 }\right)$

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

Cycle notation

$(1,3)(2,6)$

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

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

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

$(1,3)(5,7)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.