Galois orbit of Artin representations 1.2e2_3_17.8t1.1

• Label: 1.2e2_3_17.8t1.1
• Dimension: 1
• Group: $C_8$
• Conductor: $ 2^{2} \cdot 3 \cdot 17 $
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$1$
Group:	$C_8$
Conductor:	$204= 2^{2} \cdot 3 \cdot 17 $
Artin number field:	Splitting field of $f= x^{8} - 51 x^{6} + 612 x^{4} - 2295 x^{2} + 1377 $ over $\Q$
Size of Galois orbit:	4
Smallest containing permutation representation:	$C_8$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 1 + 48\cdot 89 + 32\cdot 89^{2} + 8\cdot 89^{3} + 12\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 13 + 75\cdot 89 + 49\cdot 89^{2} + 11\cdot 89^{3} + 44\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 18 + 50\cdot 89 + 48\cdot 89^{2} + 78\cdot 89^{3} + 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 25 + 51\cdot 89 + 54\cdot 89^{2} + 85\cdot 89^{3} + 33\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 64 + 37\cdot 89 + 34\cdot 89^{2} + 3\cdot 89^{3} + 55\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 71 + 38\cdot 89 + 40\cdot 89^{2} + 10\cdot 89^{3} + 87\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 76 + 13\cdot 89 + 39\cdot 89^{2} + 77\cdot 89^{3} + 44\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 88 + 40\cdot 89 + 56\cdot 89^{2} + 80\cdot 89^{3} + 76\cdot 89^{4} +O\left(89^{ 5 }\right)$

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

Cycle notation

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.