Galois orbit of Artin representations 4.2e10_3e2_5e2.8t22.4

• Label: 4.2e10_3e2_5e2.8t22.4
• 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} - 3 x^{6} + 5 x^{4} - 6 x^{2} + 4 $ 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 5.

Roots:

$r_{ 1 }$	$=$	$ 21 + 74\cdot 241 + 55\cdot 241^{2} + 148\cdot 241^{3} + 227\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 24 + 64\cdot 241 + 202\cdot 241^{2} + 35\cdot 241^{3} + 106\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 55 + 207\cdot 241 + 226\cdot 241^{2} + 184\cdot 241^{3} + 119\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 96 + 46\cdot 241 + 216\cdot 241^{2} + 117\cdot 241^{3} + 187\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 145 + 194\cdot 241 + 24\cdot 241^{2} + 123\cdot 241^{3} + 53\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 186 + 33\cdot 241 + 14\cdot 241^{2} + 56\cdot 241^{3} + 121\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 217 + 176\cdot 241 + 38\cdot 241^{2} + 205\cdot 241^{3} + 134\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 220 + 166\cdot 241 + 185\cdot 241^{2} + 92\cdot 241^{3} + 13\cdot 241^{4} +O\left(241^{ 5 }\right)$

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

Cycle notation

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

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

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.