Galois orbit of Artin representations 4.2e4_3e2_5e2_11e2.8t22.8

• Label: 4.2e4_3e2_5e2_11e2.8t22.8
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$435600= 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2} $
Artin number field:	Splitting field of $f= x^{8} + 3 x^{6} - 12 x^{5} + 8 x^{4} - 18 x^{3} + 35 x^{2} - 18 x + 5 $ 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 $\Q_{ 421 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 120 + 261\cdot 421 + 147\cdot 421^{2} + 388\cdot 421^{3} + 315\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 128 + 341\cdot 421 + 180\cdot 421^{2} + 358\cdot 421^{3} + 241\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 220 + 277\cdot 421 + 223\cdot 421^{2} + 318\cdot 421^{3} + 212\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 231 + 315\cdot 421 + 132\cdot 421^{2} + 223\cdot 421^{3} + 365\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 268 + 97\cdot 421 + 165\cdot 421^{2} + 3\cdot 421^{3} + 71\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 326 + 141\cdot 421 + 348\cdot 421^{2} + 91\cdot 421^{3} + 213\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 403 + 411\cdot 421 + 122\cdot 421^{2} + 270\cdot 421^{3} + 40\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 409 + 257\cdot 421 + 362\cdot 421^{2} + 29\cdot 421^{3} + 223\cdot 421^{4} +O\left(421^{ 5 }\right)$

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

Cycle notation

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

$(1,2)(4,8)$

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.