Galois orbit of Artin representations 1.3_17.8t1.1

• Label: 1.3_17.8t1.1
• Dimension: 1
• Group: $C_8$
• Conductor: $ 3 \cdot 17 $
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$1$
Group:	$C_8$
Conductor:	$51= 3 \cdot 17 $
Artin number field:	Splitting field of $f= x^{8} - x^{7} + 10 x^{6} - 11 x^{5} + 15 x^{4} - 61 x^{3} + 58 x^{2} - 47 x + 103 $ over $\Q$
Size of Galois orbit:	4
Smallest containing permutation representation:	$C_8$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 1 + 7\cdot 67 + 61\cdot 67^{2} + 65\cdot 67^{3} + 58\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 5 + 10\cdot 67 + 7\cdot 67^{2} + 12\cdot 67^{3} + 11\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 27 + 19\cdot 67 + 22\cdot 67^{3} + 21\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 29 + 64\cdot 67 + 42\cdot 67^{2} + 13\cdot 67^{3} + 24\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 45 + 60\cdot 67^{2} + 34\cdot 67^{3} + 17\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 50 + 38\cdot 67 + 5\cdot 67^{2} + 40\cdot 67^{3} + 17\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 51 + 38\cdot 67 + 35\cdot 67^{3} + 17\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 61 + 21\cdot 67 + 23\cdot 67^{2} + 44\cdot 67^{3} + 32\cdot 67^{4} +O\left(67^{ 5 }\right)$

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

Cycle notation

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

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

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$	$c2$	$c3$	$c4$
$1$	$1$	$()$	$1$	$1$	$1$	$1$
$1$	$2$	$(1,2)(3,7)(4,6)(5,8)$	$-1$	$-1$	$-1$	$-1$
$1$	$4$	$(1,7,2,3)(4,5,6,8)$	$\zeta_{8}^{2}$	$-\zeta_{8}^{2}$	$\zeta_{8}^{2}$	$-\zeta_{8}^{2}$
$1$	$4$	$(1,3,2,7)(4,8,6,5)$	$-\zeta_{8}^{2}$	$\zeta_{8}^{2}$	$-\zeta_{8}^{2}$	$\zeta_{8}^{2}$
$1$	$8$	$(1,5,7,6,2,8,3,4)$	$\zeta_{8}$	$\zeta_{8}^{3}$	$-\zeta_{8}$	$-\zeta_{8}^{3}$
$1$	$8$	$(1,6,3,5,2,4,7,8)$	$\zeta_{8}^{3}$	$\zeta_{8}$	$-\zeta_{8}^{3}$	$-\zeta_{8}$
$1$	$8$	$(1,8,7,4,2,5,3,6)$	$-\zeta_{8}$	$-\zeta_{8}^{3}$	$\zeta_{8}$	$\zeta_{8}^{3}$
$1$	$8$	$(1,4,3,8,2,6,7,5)$	$-\zeta_{8}^{3}$	$-\zeta_{8}$	$\zeta_{8}^{3}$	$\zeta_{8}$

The blue line marks the conjugacy class containing complex conjugation.