Galois orbit of Artin representations 2.3e2_13e2.8t17.1

• Label: 2.3e2_13e2.8t17.1
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 3^{2} \cdot 13^{2}$
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$1521= 3^{2} \cdot 13^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{6} - 3 x^{5} + 3 x^{4} + 3 x^{3} - 2 x^{2} + 1 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_4\wr C_2$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 48 + 52\cdot 139 + 37\cdot 139^{2} + 86\cdot 139^{3} + 30\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 55 + 24\cdot 139 + 70\cdot 139^{2} + 32\cdot 139^{3} + 93\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 74 + 93\cdot 139 + 61\cdot 139^{2} + 116\cdot 139^{3} + 51\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 77 + 83\cdot 139 + 15\cdot 139^{2} + 95\cdot 139^{3} + 98\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 90 + 116\cdot 139 + 20\cdot 139^{2} + 25\cdot 139^{3} + 118\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 93 + 94\cdot 139 + 130\cdot 139^{2} + 28\cdot 139^{3} + 62\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 122 + 72\cdot 139 + 124\cdot 139^{2} + 31\cdot 139^{3} + 123\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 136 + 17\cdot 139 + 95\cdot 139^{2} + 117\cdot 139^{4} +O\left(139^{ 5 }\right)$

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

Cycle notation

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

$(3,7,4,5)$

$(3,4)(5,7)$

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

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$	$c2$
$1$	$1$	$()$	$2$	$2$
$1$	$2$	$(1,2)(3,4)(5,7)(6,8)$	$-2$	$-2$
$2$	$2$	$(3,4)(5,7)$	$0$	$0$
$4$	$2$	$(1,5)(2,7)(3,8)(4,6)$	$0$	$0$
$1$	$4$	$(1,6,2,8)(3,5,4,7)$	$2 \zeta_{4}$	$-2 \zeta_{4}$
$1$	$4$	$(1,8,2,6)(3,7,4,5)$	$-2 \zeta_{4}$	$2 \zeta_{4}$
$2$	$4$	$(3,7,4,5)$	$\zeta_{4} - 1$	$-\zeta_{4} - 1$
$2$	$4$	$(3,5,4,7)$	$-\zeta_{4} - 1$	$\zeta_{4} - 1$
$2$	$4$	$(1,2)(3,5,4,7)(6,8)$	$-\zeta_{4} + 1$	$\zeta_{4} + 1$
$2$	$4$	$(1,2)(3,7,4,5)(6,8)$	$\zeta_{4} + 1$	$-\zeta_{4} + 1$
$2$	$4$	$(1,6,2,8)(3,7,4,5)$	$0$	$0$
$4$	$4$	$(1,7,2,5)(3,8,4,6)$	$0$	$0$
$4$	$8$	$(1,3,8,7,2,4,6,5)$	$0$	$0$
$4$	$8$	$(1,7,6,3,2,5,8,4)$	$0$	$0$

The blue line marks the conjugacy class containing complex conjugation.