Galois orbit of Artin representations 1.41.8t1.1

• Label: 1.41.8t1.1
• Dimension: 1
• Group: $C_8$
• Conductor: $ 41 $
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$1$
Group:	$C_8$
Conductor:	$41 $
Artin number field:	Splitting field of $f= x^{8} - x^{7} + 3 x^{6} - 11 x^{5} + 44 x^{4} + 53 x^{3} + 153 x^{2} + 160 x + 59 $ 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 $\Q_{ 59 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 7\cdot 59 + 43\cdot 59^{2} + 36\cdot 59^{3} + 7\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 18 + 7\cdot 59 + 53\cdot 59^{2} + 56\cdot 59^{3} + 57\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 32 + 15\cdot 59 + 17\cdot 59^{2} + 49\cdot 59^{3} + 48\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 37 + 52\cdot 59 + 17\cdot 59^{2} + 4\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 45 + 3\cdot 59 + 54\cdot 59^{2} + 8\cdot 59^{3} + 4\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 50 + 42\cdot 59 + 8\cdot 59^{2} + 27\cdot 59^{3} + 35\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 56 + 43\cdot 59 + 41\cdot 59^{2} + 38\cdot 59^{3} + 35\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 58 + 3\cdot 59 + 18\cdot 59^{3} + 42\cdot 59^{4} +O\left(59^{ 5 }\right)$

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

Cycle notation

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.