Galois orbit of Artin representations 1.113.8t1.1

• Label: 1.113.8t1.1
• Dimension: 1
• Group: $C_8$
• Conductor: $ 113 $
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$1$
Group:	$C_8$
Conductor:	$113 $
Artin number field:	Splitting field of $f= x^{8} - x^{7} - 49 x^{6} - 16 x^{5} + 511 x^{4} + 367 x^{3} - 1499 x^{2} - 798 x + 1372 $ 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 $\Q_{ 97 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 3 + 80\cdot 97 + 37\cdot 97^{2} + 72\cdot 97^{3} + 43\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 10 + 34\cdot 97 + 60\cdot 97^{2} + 67\cdot 97^{3} + 2\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 27 + 82\cdot 97 + 89\cdot 97^{2} + 44\cdot 97^{3} + 13\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 30 + 40\cdot 97 + 44\cdot 97^{2} + 30\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 31 + 39\cdot 97 + 24\cdot 97^{2} + 23\cdot 97^{3} + 76\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 55 + 42\cdot 97 + 38\cdot 97^{2} + 11\cdot 97^{3} + 86\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 59 + 74\cdot 97 + 29\cdot 97^{2} + 84\cdot 97^{3} + 38\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 77 + 91\cdot 97 + 62\cdot 97^{2} + 83\cdot 97^{3} + 96\cdot 97^{4} +O\left(97^{ 5 }\right)$

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

Cycle notation

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.