Galois orbit of Artin representations 2.3e2_13e2.8t17.2

• Label: 2.3e2_13e2.8t17.2
• 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} - 3 x^{7} - 5 x^{6} + 15 x^{5} + 15 x^{4} - 6 x^{3} - 71 x^{2} - 36 x + 133 $ 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 6.

Roots:

$r_{ 1 }$	$=$	$ 11 + 84\cdot 139 + 35\cdot 139^{2} + 110\cdot 139^{3} + 86\cdot 139^{4} + 23\cdot 139^{5} +O\left(139^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 17 + 136\cdot 139 + 63\cdot 139^{2} + 65\cdot 139^{3} + 60\cdot 139^{4} + 108\cdot 139^{5} +O\left(139^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 19 + 5\cdot 139 + 33\cdot 139^{2} + 38\cdot 139^{3} + 93\cdot 139^{4} + 32\cdot 139^{5} +O\left(139^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 41 + 12\cdot 139 + 105\cdot 139^{2} + 122\cdot 139^{3} + 111\cdot 139^{4} + 112\cdot 139^{5} +O\left(139^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 98 + 41\cdot 139 + 20\cdot 139^{2} + 106\cdot 139^{3} + 127\cdot 139^{4} + 94\cdot 139^{5} +O\left(139^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 114 + 89\cdot 139 + 45\cdot 139^{2} + 44\cdot 139^{3} + 6\cdot 139^{4} + 97\cdot 139^{5} +O\left(139^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 123 + 42\cdot 139 + 104\cdot 139^{2} + 59\cdot 139^{3} + 6\cdot 139^{4} + 64\cdot 139^{5} +O\left(139^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 136 + 4\cdot 139 + 9\cdot 139^{2} + 9\cdot 139^{3} + 63\cdot 139^{4} + 22\cdot 139^{5} +O\left(139^{ 6 }\right)$

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

Cycle notation

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

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

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

$(1,6,4,2)$

$(1,4)(2,6)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.