Galois orbit of Artin representations 4.2e10_5e2_7e2.8t22.6

• Label: 4.2e10_5e2_7e2.8t22.6
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{10} \cdot 5^{2} \cdot 7^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$1254400= 2^{10} \cdot 5^{2} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{7} - 2 x^{6} + 16 x^{5} + 8 x^{4} - 28 x^{3} - 16 x^{2} + 24 x + 18 $ 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 $\Q_{ 569 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 15 + 379\cdot 569 + 346\cdot 569^{2} + 168\cdot 569^{3} + 184\cdot 569^{4} +O\left(569^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 20 + 188\cdot 569 + 534\cdot 569^{2} + 123\cdot 569^{3} + 356\cdot 569^{4} +O\left(569^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 62 + 115\cdot 569 + 250\cdot 569^{2} + 211\cdot 569^{3} + 103\cdot 569^{4} +O\left(569^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 79 + 100\cdot 569 + 53\cdot 569^{2} + 318\cdot 569^{3} + 147\cdot 569^{4} +O\left(569^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 214 + 270\cdot 569 + 152\cdot 569^{2} + 40\cdot 569^{3} + 20\cdot 569^{4} +O\left(569^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 286 + 427\cdot 569 + 280\cdot 569^{2} + 264\cdot 569^{3} + 480\cdot 569^{4} +O\left(569^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 474 + 455\cdot 569 + 6\cdot 569^{2} + 65\cdot 569^{3} + 494\cdot 569^{4} +O\left(569^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 561 + 339\cdot 569 + 82\cdot 569^{2} + 515\cdot 569^{3} + 489\cdot 569^{4} +O\left(569^{ 5 }\right)$

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

Cycle notation

$(1,3)(6,8)$

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

$(1,3)(2,7)$

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

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

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

$(1,3)(4,5)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.