Galois orbit of Artin representations 4.2e10_3e2_5e2.8t22.2

• Label: 4.2e10_3e2_5e2.8t22.2
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{10} \cdot 3^{2} \cdot 5^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$230400= 2^{10} \cdot 3^{2} \cdot 5^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{6} - 8 x^{5} - 5 x^{4} + 8 x^{3} + 12 x^{2} + 4 x - 1 $ 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 $\Q_{ 241 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 12 + 52\cdot 241 + 101\cdot 241^{2} + 200\cdot 241^{3} + 174\cdot 241^{4} + 38\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 37 + 185\cdot 241 + 115\cdot 241^{2} + 90\cdot 241^{3} + 222\cdot 241^{4} + 152\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 47 + 218\cdot 241 + 99\cdot 241^{2} + 83\cdot 241^{3} + 60\cdot 241^{4} + 164\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 95 + 209\cdot 241 + 59\cdot 241^{2} + 96\cdot 241^{3} + 145\cdot 241^{4} + 203\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 124 + 42\cdot 241 + 233\cdot 241^{2} + 3\cdot 241^{4} + 7\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 207 + 199\cdot 241 + 191\cdot 241^{2} + 137\cdot 241^{3} + 214\cdot 241^{4} + 171\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 216 + 11\cdot 241 + 89\cdot 241^{2} + 60\cdot 241^{3} + 32\cdot 241^{4} + 107\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 226 + 44\cdot 241 + 73\cdot 241^{2} + 53\cdot 241^{3} + 111\cdot 241^{4} + 118\cdot 241^{5} +O\left(241^{ 6 }\right)$

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

Cycle notation

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

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

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.