Galois orbit of Artin representations 4.2e4_3e2_7e2_11e2.8t22.5

• Label: 4.2e4_3e2_7e2_11e2.8t22.5
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$853776= 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{7} + 9 x^{6} - 4 x^{5} - 16 x^{4} + 34 x^{3} - 20 x^{2} - 24 x + 37 $ 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 $\Q_{ 37 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 17\cdot 37 + 10\cdot 37^{2} + 30\cdot 37^{3} + 9\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 7 + 8\cdot 37 + 14\cdot 37^{2} + 8\cdot 37^{3} + 25\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 9 + 7\cdot 37 + 28\cdot 37^{2} + 35\cdot 37^{3} + 13\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 14 + 3\cdot 37 + 21\cdot 37^{2} + 9\cdot 37^{3} + 32\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 16 + 9\cdot 37 + 14\cdot 37^{2} + 35\cdot 37^{3} + 17\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 17 + 37 + 28\cdot 37^{2} + 20\cdot 37^{3} + 21\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 22 + 19\cdot 37 + 11\cdot 37^{2} + 12\cdot 37^{3} + 36\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 30 + 7\cdot 37 + 20\cdot 37^{2} + 32\cdot 37^{3} + 27\cdot 37^{4} +O\left(37^{ 5 }\right)$

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

Cycle notation

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

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

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

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

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

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

$(4,5)(7,8)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.