Galois orbit of Artin representations 2.3_97.8t17.1

• Label: 2.3_97.8t17.1
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 3 \cdot 97 $
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$291= 3 \cdot 97 $
Artin number field:	Splitting field of $f= x^{8} - x^{7} + 3 x^{6} + 6 x^{5} + 3 x^{4} + 8 x^{3} + 20 x^{2} + 8 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 $\Q_{ 103 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 2 + 94\cdot 103 + 95\cdot 103^{2} + 35\cdot 103^{3} + 24\cdot 103^{4} + 47\cdot 103^{5} +O\left(103^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 6 + 70\cdot 103 + 94\cdot 103^{2} + 38\cdot 103^{3} + 3\cdot 103^{4} + 2\cdot 103^{5} +O\left(103^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 10 + 76\cdot 103 + 49\cdot 103^{2} + 86\cdot 103^{3} + 39\cdot 103^{4} + 9\cdot 103^{5} +O\left(103^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 11 + 69\cdot 103 + 36\cdot 103^{2} + 41\cdot 103^{3} + 81\cdot 103^{4} + 47\cdot 103^{5} +O\left(103^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 50 + 62\cdot 103 + 59\cdot 103^{2} + 62\cdot 103^{3} + 62\cdot 103^{4} + 75\cdot 103^{5} +O\left(103^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 54 + 13\cdot 103 + 34\cdot 103^{2} + 54\cdot 103^{3} + 83\cdot 103^{4} + 79\cdot 103^{5} +O\left(103^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 79 + 86\cdot 103 + 18\cdot 103^{2} + 100\cdot 103^{3} + 72\cdot 103^{4} + 51\cdot 103^{5} +O\left(103^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 98 + 42\cdot 103 + 22\cdot 103^{2} + 95\cdot 103^{3} + 43\cdot 103^{4} + 98\cdot 103^{5} +O\left(103^{ 6 }\right)$

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

Cycle notation

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

$(2,6)(4,7)$

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

$(2,7,6,4)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.