Galois orbit of Artin representations 2.2e4_7.8t17.1

• Label: 2.2e4_7.8t17.1
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 2^{4} \cdot 7 $
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$112= 2^{4} \cdot 7 $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} + 6 x^{6} - 8 x^{5} + 10 x^{4} - 9 x^{3} + 6 x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_4\wr C_2$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 239 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 30 + 93\cdot 239 + 155\cdot 239^{2} + 152\cdot 239^{3} + 154\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 92 + 154\cdot 239 + 26\cdot 239^{2} + 233\cdot 239^{3} + 21\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 122 + 203\cdot 239 + 211\cdot 239^{2} + 50\cdot 239^{3} + 166\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 126 + 68\cdot 239 + 25\cdot 239^{2} + 86\cdot 239^{3} + 60\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 187 + 73\cdot 239 + 178\cdot 239^{2} + 232\cdot 239^{3} + 208\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 194 + 202\cdot 239 + 87\cdot 239^{2} + 65\cdot 239^{3} + 69\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 211 + 85\cdot 239 + 207\cdot 239^{2} + 116\cdot 239^{3} + 137\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 236 + 73\cdot 239 + 63\cdot 239^{2} + 18\cdot 239^{3} + 137\cdot 239^{4} +O\left(239^{ 5 }\right)$

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

Cycle notation

$(3,5)(4,7)$

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

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

$(3,7,5,4)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.