Galois orbit of Artin representations 4.2e4_3e4_5e2_13e2.8t22.6

• Label: 4.2e4_3e4_5e2_13e2.8t22.6
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 13^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$5475600= 2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 13^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{7} + 5 x^{6} + 8 x^{5} - 41 x^{4} + 70 x^{3} - 22 x^{2} - 164 x + 244 $ 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 $\Q_{ 61 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 3\cdot 61 + 59\cdot 61^{2} + 30\cdot 61^{3} + 28\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 3 + 46\cdot 61 + 49\cdot 61^{2} + 5\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 7 + 29\cdot 61 + 41\cdot 61^{2} + 61^{3} + 50\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 20 + 51\cdot 61 + 41\cdot 61^{2} + 4\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 21 + 55\cdot 61 + 42\cdot 61^{2} + 39\cdot 61^{3} + 52\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 22 + 12\cdot 61 + 39\cdot 61^{2} + 50\cdot 61^{3} + 36\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 23 + 5\cdot 61 + 57\cdot 61^{2} + 10\cdot 61^{3} + 57\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 30 + 41\cdot 61 + 34\cdot 61^{2} + 47\cdot 61^{3} + 9\cdot 61^{4} +O\left(61^{ 5 }\right)$

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

Cycle notation

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

$(1,6)(3,7)$

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

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

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

$(1,6)(2,8)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.