Galois orbit of Artin representations 4.2e6_3e2_5e2_7e2.8t22.10

• Label: 4.2e6_3e2_5e2_7e2.8t22.10
• 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} + 10 x^{6} - 15 x^{5} + 28 x^{4} - 34 x^{3} + 30 x^{2} - 21 x + 7 $ 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 $\Q_{ 211 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 26 + 93\cdot 211 + 85\cdot 211^{2} + 17\cdot 211^{3} + 151\cdot 211^{4} + 138\cdot 211^{5} +O\left(211^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 49 + 114\cdot 211 + 176\cdot 211^{2} + 2\cdot 211^{3} + 147\cdot 211^{4} + 131\cdot 211^{5} +O\left(211^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 61 + 71\cdot 211 + 72\cdot 211^{2} + 26\cdot 211^{3} + 101\cdot 211^{4} + 44\cdot 211^{5} +O\left(211^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 83 + 160\cdot 211 + 122\cdot 211^{2} + 80\cdot 211^{3} + 14\cdot 211^{4} + 197\cdot 211^{5} +O\left(211^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 130 + 67\cdot 211 + 167\cdot 211^{2} + 26\cdot 211^{3} + 62\cdot 211^{4} + 198\cdot 211^{5} +O\left(211^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 149 + 122\cdot 211 + 59\cdot 211^{2} + 77\cdot 211^{3} + 33\cdot 211^{4} + 193\cdot 211^{5} +O\left(211^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 165 + 45\cdot 211 + 154\cdot 211^{2} + 35\cdot 211^{3} + 12\cdot 211^{4} + 104\cdot 211^{5} +O\left(211^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 183 + 168\cdot 211 + 5\cdot 211^{2} + 155\cdot 211^{3} + 111\cdot 211^{4} + 47\cdot 211^{5} +O\left(211^{ 6 }\right)$

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

Cycle notation

$(2,8)(4,6)$

$(2,8)(3,5)$

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

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

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

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

$(1,7)(2,8)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.