Galois orbit of Artin representations 2.5_29e2.8t17.1

• Label: 2.5_29e2.8t17.1
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 5 \cdot 29^{2}$
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$4205= 5 \cdot 29^{2} $
Artin number field:	Splitting field of $f= x^{8} - x^{7} - 3 x^{6} + 3 x^{5} + 3 x^{4} - 6 x^{3} - 2 x^{2} + 3 x + 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:

$r_{ 1 }$	$=$	$ 10 + 35\cdot 139 + 67\cdot 139^{2} + 29\cdot 139^{3} + 61\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 46 + 33\cdot 139 + 70\cdot 139^{2} + 108\cdot 139^{3} + 130\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 47 + 91\cdot 139 + 134\cdot 139^{2} + 48\cdot 139^{3} + 84\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 50 + 39\cdot 139 + 67\cdot 139^{2} + 129\cdot 139^{3} + 105\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 51 + 11\cdot 139 + 3\cdot 139^{2} + 122\cdot 139^{3} + 53\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 56 + 69\cdot 139 + 27\cdot 139^{2} + 127\cdot 139^{3} + 122\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 62 + 62\cdot 139 + 133\cdot 139^{2} + 25\cdot 139^{3} + 12\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 96 + 74\cdot 139 + 52\cdot 139^{2} + 103\cdot 139^{3} + 123\cdot 139^{4} +O\left(139^{ 5 }\right)$

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

Cycle notation

$(2,5,4,6)$

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

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

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

$(2,4)(5,6)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.