Galois orbit of Artin representations 4.3e3_7e2_61e2.8t29.2

• Label: 4.3e3_7e2_61e2.8t29.2
• Dimension: 4
• Group: $(((C_4 \times C_2): C_2):C_2):C_2$
• Conductor: $ 3^{3} \cdot 7^{2} \cdot 61^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(((C_4 \times C_2): C_2):C_2):C_2$
Conductor:	$4922883= 3^{3} \cdot 7^{2} \cdot 61^{2} $
Artin number field:	Splitting field of $f= x^{8} - x^{7} - 9 x^{6} + 4 x^{5} + 29 x^{4} - 6 x^{3} - 24 x^{2} + 6 x - 3 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(((C_4 \times C_2): C_2):C_2):C_2$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 1033 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 13 + 677\cdot 1033 + 62\cdot 1033^{2} + 734\cdot 1033^{3} + 780\cdot 1033^{4} + 493\cdot 1033^{5} + 192\cdot 1033^{6} +O\left(1033^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 89 + 103\cdot 1033 + 238\cdot 1033^{2} + 820\cdot 1033^{3} + 748\cdot 1033^{4} + 916\cdot 1033^{5} + 707\cdot 1033^{6} +O\left(1033^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 144 + 1031\cdot 1033 + 113\cdot 1033^{2} + 148\cdot 1033^{3} + 643\cdot 1033^{4} + 285\cdot 1033^{5} + 1017\cdot 1033^{6} +O\left(1033^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 205 + 390\cdot 1033 + 138\cdot 1033^{2} + 946\cdot 1033^{3} + 215\cdot 1033^{4} + 966\cdot 1033^{5} + 781\cdot 1033^{6} +O\left(1033^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 416 + 543\cdot 1033 + 749\cdot 1033^{2} + 164\cdot 1033^{3} + 603\cdot 1033^{4} + 784\cdot 1033^{5} + 226\cdot 1033^{6} +O\left(1033^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 557 + 954\cdot 1033 + 448\cdot 1033^{2} + 527\cdot 1033^{3} + 181\cdot 1033^{4} + 797\cdot 1033^{5} + 758\cdot 1033^{6} +O\left(1033^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 799 + 820\cdot 1033 + 99\cdot 1033^{2} + 12\cdot 1033^{3} + 832\cdot 1033^{4} + 665\cdot 1033^{5} + 293\cdot 1033^{6} +O\left(1033^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 877 + 644\cdot 1033 + 214\cdot 1033^{2} + 779\cdot 1033^{3} + 126\cdot 1033^{4} + 255\cdot 1033^{5} + 153\cdot 1033^{6} +O\left(1033^{ 7 }\right)$

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

Cycle notation

$(2,4)(3,5)$

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

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

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

$(1,6)(7,8)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.