Galois orbit of Artin representations 2.2e2_3e2_17e2.8t17.1

• Label: 2.2e2_3e2_17e2.8t17.1
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 2^{2} \cdot 3^{2} \cdot 17^{2}$
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$10404= 2^{2} \cdot 3^{2} \cdot 17^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + 4 x^{6} - 8 x^{4} + 8 x^{3} + 15 x^{2} + 6 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 $\Q_{ 157 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 7 + 91\cdot 157 + 64\cdot 157^{2} + 155\cdot 157^{3} + 23\cdot 157^{4} + 20\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 15 + 128\cdot 157 + 78\cdot 157^{2} + 95\cdot 157^{3} + 57\cdot 157^{4} + 89\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 33 + 153\cdot 157 + 109\cdot 157^{2} + 61\cdot 157^{3} + 143\cdot 157^{4} + 80\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 84 + 98\cdot 157 + 154\cdot 157^{2} + 22\cdot 157^{3} + 31\cdot 157^{4} + 77\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 93 + 57\cdot 157 + 28\cdot 157^{2} + 98\cdot 157^{3} + 15\cdot 157^{4} + 122\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 125 + 148\cdot 157 + 51\cdot 157^{2} + 34\cdot 157^{3} + 134\cdot 157^{4} + 70\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 131 + 11\cdot 157 + 88\cdot 157^{2} + 115\cdot 157^{3} + 87\cdot 157^{4} + 33\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 142 + 95\cdot 157 + 51\cdot 157^{2} + 44\cdot 157^{3} + 134\cdot 157^{4} + 133\cdot 157^{5} +O\left(157^{ 6 }\right)$

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

Cycle notation

$(4,8,6,5)$

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

$(4,6)(5,8)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.