Galois orbit of Artin representations 4.2e4_3e2_5e2_13e2.8t22.4

• Label: 4.2e4_3e2_5e2_13e2.8t22.4
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$608400= 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} - 6 x^{6} + 12 x^{5} + 24 x^{4} - 14 x^{3} - 31 x^{2} + 4 x + 13 $ 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:

$r_{ 1 }$	$=$	$ 8 + 34\cdot 61 + 55\cdot 61^{2} + 46\cdot 61^{3} + 51\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 11 + 26\cdot 61 + 8\cdot 61^{2} + 35\cdot 61^{3} + 50\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 17 + 58\cdot 61 + 60\cdot 61^{2} + 4\cdot 61^{3} + 14\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 28 + 59\cdot 61 + 42\cdot 61^{2} + 14\cdot 61^{3} + 18\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 37 + 14\cdot 61 + 44\cdot 61^{2} + 61^{3} + 16\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 45 + 9\cdot 61 + 61^{2} + 45\cdot 61^{3} + 13\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 49 + 8\cdot 61 + 35\cdot 61^{2} + 31\cdot 61^{3} + 54\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 51 + 32\cdot 61 + 56\cdot 61^{2} + 2\cdot 61^{3} + 25\cdot 61^{4} +O\left(61^{ 5 }\right)$

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

Cycle notation

$(4,7)(6,8)$

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

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.