Galois orbit of Artin representations 2.2e3_17e2.8t7.2

• Label: 2.2e3_17e2.8t7.2
• Dimension: 2
• Group: $C_8:C_2$
• Conductor: $ 2^{3} \cdot 17^{2}$
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_8:C_2$
Conductor:	$2312= 2^{3} \cdot 17^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + 6 x^{6} - 3 x^{5} + 19 x^{4} - 14 x^{3} + 6 x^{2} + 36 x + 52 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_8:C_2$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 239 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 19 + 103\cdot 239 + 135\cdot 239^{2} + 94\cdot 239^{3} + 168\cdot 239^{4} + 214\cdot 239^{5} +O\left(239^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 25 + 22\cdot 239 + 177\cdot 239^{2} + 211\cdot 239^{3} + 20\cdot 239^{4} + 206\cdot 239^{5} +O\left(239^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 87 + 41\cdot 239 + 15\cdot 239^{2} + 178\cdot 239^{3} + 223\cdot 239^{4} + 176\cdot 239^{5} +O\left(239^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 93 + 199\cdot 239 + 56\cdot 239^{2} + 56\cdot 239^{3} + 76\cdot 239^{4} + 168\cdot 239^{5} +O\left(239^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 149 + 57\cdot 239 + 11\cdot 239^{2} + 112\cdot 239^{3} + 46\cdot 239^{4} + 118\cdot 239^{5} +O\left(239^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 176 + 40\cdot 239 + 239^{2} + 208\cdot 239^{3} + 26\cdot 239^{4} + 51\cdot 239^{5} +O\left(239^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 191 + 134\cdot 239 + 45\cdot 239^{2} + 119\cdot 239^{3} + 206\cdot 239^{4} + 43\cdot 239^{5} +O\left(239^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 218 + 117\cdot 239 + 35\cdot 239^{2} + 215\cdot 239^{3} + 186\cdot 239^{4} + 215\cdot 239^{5} +O\left(239^{ 6 }\right)$

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

Cycle notation

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

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

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

$(5,8)(6,7)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.