Galois orbit of Artin representations 4.2e4_3e2_7e2_13e2.8t22.9

• Label: 4.2e4_3e2_7e2_13e2.8t22.9
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$1192464= 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{7} - x^{6} + 16 x^{5} + x^{4} - 24 x^{3} - 12 x^{2} + 36 $ 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 $\Q_{ 673 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 70 + 34\cdot 673 + 139\cdot 673^{2} + 260\cdot 673^{3} + 4\cdot 673^{4} +O\left(673^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 102 + 366\cdot 673 + 242\cdot 673^{2} + 99\cdot 673^{3} + 345\cdot 673^{4} +O\left(673^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 107 + 424\cdot 673 + 74\cdot 673^{2} + 172\cdot 673^{3} + 656\cdot 673^{4} +O\left(673^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 291 + 93\cdot 673 + 646\cdot 673^{2} + 593\cdot 673^{3} + 313\cdot 673^{4} +O\left(673^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 425 + 425\cdot 673 + 467\cdot 673^{2} + 635\cdot 673^{3} + 199\cdot 673^{4} +O\left(673^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 525 + 402\cdot 673 + 157\cdot 673^{2} + 617\cdot 673^{3} + 175\cdot 673^{4} +O\left(673^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 556 + 7\cdot 673 + 159\cdot 673^{2} + 462\cdot 673^{3} + 337\cdot 673^{4} +O\left(673^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 620 + 264\cdot 673 + 132\cdot 673^{2} + 524\cdot 673^{3} + 658\cdot 673^{4} +O\left(673^{ 5 }\right)$

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

Cycle notation

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

$(2,7)(4,5)$

$(3,6)(4,5)$

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

$(1,8)(4,5)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.