Galois orbit of Artin representations 4.2e10_3e2_11e2.8t22.4

• Label: 4.2e10_3e2_11e2.8t22.4
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{10} \cdot 3^{2} \cdot 11^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$1115136= 2^{10} \cdot 3^{2} \cdot 11^{2} $
Artin number field:	Splitting field of $f= x^{8} + 4 x^{6} - 8 x^{5} + 5 x^{4} - 16 x^{3} + 18 x^{2} - 4 x + 7 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3 : D_4 $
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 3 + 33\cdot 97 + 26\cdot 97^{2} + 52\cdot 97^{3} + 58\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 34 + 85\cdot 97 + 8\cdot 97^{2} + 43\cdot 97^{3} + 66\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 40 + 41\cdot 97 + 78\cdot 97^{2} + 61\cdot 97^{3} + 38\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 52 + 94\cdot 97 + 52\cdot 97^{2} + 59\cdot 97^{3} + 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 53 + 43\cdot 97 + 16\cdot 97^{2} + 40\cdot 97^{3} + 81\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 63 + 7\cdot 97 + 27\cdot 97^{2} + 40\cdot 97^{3} + 56\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 68 + 69\cdot 97 + 53\cdot 97^{2} + 29\cdot 97^{3} + 87\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 75 + 12\cdot 97 + 27\cdot 97^{2} + 61\cdot 97^{3} + 94\cdot 97^{4} +O\left(97^{ 5 }\right)$

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

Cycle notation

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

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

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.