Galois orbit of Artin representations 4.2e6_3e2_5e2_7e2.8t22.13

• Label: 4.2e6_3e2_5e2_7e2.8t22.13
• 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} - x^{7} + x^{6} - x^{5} + 4 x^{4} + x^{3} + 3 x^{2} + 10 x + 11 $ 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 5.

Roots:

$r_{ 1 }$	$=$	$ 36 + 141\cdot 211 + 191\cdot 211^{2} + 138\cdot 211^{3} + 187\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 51 + 111\cdot 211 + 128\cdot 211^{2} + 142\cdot 211^{3} + 85\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 75 + 60\cdot 211 + 17\cdot 211^{2} + 116\cdot 211^{3} + 68\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 81 + 86\cdot 211 + 91\cdot 211^{2} + 179\cdot 211^{3} + 168\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 109 + 126\cdot 211 + 26\cdot 211^{2} + 193\cdot 211^{3} + 163\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 121 + 211 + 114\cdot 211^{2} + 129\cdot 211^{3} + 32\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 182 + 21\cdot 211 + 4\cdot 211^{2} + 27\cdot 211^{3} + 165\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 190 + 83\cdot 211 + 59\cdot 211^{2} + 128\cdot 211^{3} + 182\cdot 211^{4} +O\left(211^{ 5 }\right)$

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

Cycle notation

$(1,7)(2,6)$

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

$(1,7)(4,8)$

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

$(1,7)(3,5)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.