Galois orbit of Artin representations 4.5e4_11e2_31e2.8t16.7

• Label: 4.5e4_11e2_31e2.8t16.7
• Dimension: 4
• Group: $(C_8:C_2):C_2$
• Conductor: $ 5^{4} \cdot 11^{2} \cdot 31^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(C_8:C_2):C_2$
Conductor:	$72675625= 5^{4} \cdot 11^{2} \cdot 31^{2} $
Artin number field:	Splitting field of $f= x^{8} - x^{7} - 23 x^{6} + 27 x^{5} + 225 x^{4} - 212 x^{3} - 1048 x^{2} + 701 x + 2321 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(C_8:C_2):C_2$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 101 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 15 + 23\cdot 101 + 29\cdot 101^{2} + 43\cdot 101^{3} + 72\cdot 101^{4} + 47\cdot 101^{5} + 10\cdot 101^{6} +O\left(101^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 26 + 15\cdot 101 + 40\cdot 101^{2} + 81\cdot 101^{3} + 69\cdot 101^{4} + 51\cdot 101^{5} + 99\cdot 101^{6} +O\left(101^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 39 + 67\cdot 101 + 89\cdot 101^{2} + 4\cdot 101^{3} + 7\cdot 101^{4} + 24\cdot 101^{5} + 79\cdot 101^{6} +O\left(101^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 48 + 27\cdot 101 + 25\cdot 101^{2} + 37\cdot 101^{3} + 27\cdot 101^{4} + 79\cdot 101^{5} + 4\cdot 101^{6} +O\left(101^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 54 + 89\cdot 101 + 90\cdot 101^{2} + 93\cdot 101^{3} + 55\cdot 101^{4} + 100\cdot 101^{5} + 95\cdot 101^{6} +O\left(101^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 64 + 15\cdot 101 + 44\cdot 101^{2} + 89\cdot 101^{3} + 66\cdot 101^{4} + 94\cdot 101^{5} + 11\cdot 101^{6} +O\left(101^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 67 + 35\cdot 101 + 13\cdot 101^{2} + 22\cdot 101^{3} + 42\cdot 101^{4} + 45\cdot 101^{5} + 90\cdot 101^{6} +O\left(101^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 92 + 28\cdot 101 + 71\cdot 101^{2} + 31\cdot 101^{3} + 62\cdot 101^{4} + 61\cdot 101^{5} + 11\cdot 101^{6} +O\left(101^{ 7 }\right)$

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

Cycle notation

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

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

$(2,3)(5,6)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.