Galois orbit of Artin representations 4.2e4_3e2_73.8t29.4

• Label: 4.2e4_3e2_73.8t29.4
• Dimension: 4
• Group: $(((C_4 \times C_2): C_2):C_2):C_2$
• Conductor: $ 2^{4} \cdot 3^{2} \cdot 73 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(((C_4 \times C_2): C_2):C_2):C_2$
Conductor:	$10512= 2^{4} \cdot 3^{2} \cdot 73 $
Artin number field:	Splitting field of $f= x^{8} - x^{6} - 2 x^{5} - x^{4} + 2 x^{3} + 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:

$r_{ 1 }$	$=$	$ 27 + 419\cdot 733 + 41\cdot 733^{2} + 685\cdot 733^{3} + 301\cdot 733^{4} + 549\cdot 733^{5} +O\left(733^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 33 + 263\cdot 733 + 550\cdot 733^{2} + 676\cdot 733^{3} + 249\cdot 733^{4} + 633\cdot 733^{5} +O\left(733^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 81 + 298\cdot 733 + 660\cdot 733^{2} + 599\cdot 733^{3} + 390\cdot 733^{4} + 704\cdot 733^{5} +O\left(733^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 173 + 476\cdot 733 + 365\cdot 733^{2} + 254\cdot 733^{3} + 210\cdot 733^{4} + 267\cdot 733^{5} +O\left(733^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 341 + 44\cdot 733 + 428\cdot 733^{2} + 133\cdot 733^{3} + 456\cdot 733^{4} + 644\cdot 733^{5} +O\left(733^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 367 + 360\cdot 733 + 317\cdot 733^{2} + 178\cdot 733^{3} + 352\cdot 733^{4} + 702\cdot 733^{5} +O\left(733^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 566 + 729\cdot 733 + 336\cdot 733^{2} + 540\cdot 733^{3} + 57\cdot 733^{4} + 521\cdot 733^{5} +O\left(733^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 611 + 340\cdot 733 + 231\cdot 733^{2} + 596\cdot 733^{3} + 179\cdot 733^{4} + 375\cdot 733^{5} +O\left(733^{ 6 }\right)$

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

Cycle notation

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

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

$(1,6)(2,7)$

$(1,6)(3,8)$

$(1,6)(4,5)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.