Galois orbit of Artin representations 4.2e6_3e2_5e2_7e2.8t22.6

• Label: 4.2e6_3e2_5e2_7e2.8t22.6
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$705600= 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} - 3 x^{6} - 4 x^{5} + 3 x^{4} + 28 x^{3} + 37 x^{2} + 18 x + 3 $ 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 $\Q_{ 331 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 18 + 81\cdot 331 + 78\cdot 331^{2} + 58\cdot 331^{3} + 139\cdot 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 50 + 99\cdot 331 + 312\cdot 331^{2} + 51\cdot 331^{3} + 323\cdot 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 189 + 58\cdot 331 + 59\cdot 331^{2} + 180\cdot 331^{3} + 266\cdot 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 257 + 41\cdot 331 + 239\cdot 331^{2} + 214\cdot 331^{3} + 162\cdot 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 258 + 300\cdot 331 + 170\cdot 331^{2} + 55\cdot 331^{3} + 250\cdot 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 262 + 275\cdot 331 + 331^{2} + 146\cdot 331^{3} + 204\cdot 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 305 + 331 + 243\cdot 331^{2} + 280\cdot 331^{3} + 157\cdot 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 318 + 133\cdot 331 + 219\cdot 331^{2} + 5\cdot 331^{3} + 151\cdot 331^{4} +O\left(331^{ 5 }\right)$

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

Cycle notation

$(1,6)(3,4)$

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

$(1,6)(2,5)$

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

$(1,6)(7,8)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.