Galois orbit of Artin representations 2.3_7_13e2.8t17.1

• Label: 2.3_7_13e2.8t17.1
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 3 \cdot 7 \cdot 13^{2}$
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$3549= 3 \cdot 7 \cdot 13^{2} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} - x^{6} + 11 x^{5} - 6 x^{4} - 10 x^{3} + 9 x^{2} - 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 $\Q_{ 181 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 3 + 22\cdot 181 + 119\cdot 181^{2} + 127\cdot 181^{3} + 146\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 10 + 109\cdot 181 + 174\cdot 181^{2} + 179\cdot 181^{3} + 64\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 36 + 155\cdot 181 + 70\cdot 181^{2} + 151\cdot 181^{3} + 124\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 42 + 145\cdot 181 + 181^{2} + 36\cdot 181^{3} + 118\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 67 + 89\cdot 181 + 109\cdot 181^{2} + 58\cdot 181^{3} + 121\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 119 + 103\cdot 181 + 47\cdot 181^{2} + 136\cdot 181^{3} + 85\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 127 + 82\cdot 181 + 52\cdot 181^{2} + 106\cdot 181^{3} + 157\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 142 + 16\cdot 181 + 148\cdot 181^{2} + 108\cdot 181^{3} + 85\cdot 181^{4} +O\left(181^{ 5 }\right)$

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

Cycle notation

$(1,5)(4,6)$

$(1,4,5,6)$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.