Galois orbit of Artin representations 2.3_73.8t17.1

• Label: 2.3_73.8t17.1
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 3 \cdot 73 $
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$219= 3 \cdot 73 $
Artin number field:	Splitting field of $f= x^{8} + 2 x^{6} - 2 x^{5} - 6 x^{4} - 2 x^{3} + 9 x^{2} + 10 x + 4 $ 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 $\Q_{ 223 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 15 + 197\cdot 223 + 5\cdot 223^{2} + 102\cdot 223^{3} + 20\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 21 + 172\cdot 223 + 66\cdot 223^{2} + 48\cdot 223^{3} + 21\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 28 + 198\cdot 223 + 46\cdot 223^{2} + 47\cdot 223^{3} + 85\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 44 + 109\cdot 223 + 187\cdot 223^{2} + 153\cdot 223^{3} + 131\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 71 + 129\cdot 223 + 21\cdot 223^{2} + 213\cdot 223^{3} + 207\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 133 + 204\cdot 223 + 210\cdot 223^{2} + 142\cdot 223^{3} + 100\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 175 + 160\cdot 223 + 209\cdot 223^{2} + 198\cdot 223^{3} + 189\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 182 + 166\cdot 223 + 142\cdot 223^{2} + 208\cdot 223^{3} + 134\cdot 223^{4} +O\left(223^{ 5 }\right)$

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

Cycle notation

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

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

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

$(2,5)(3,8)$

$(2,3,5,8)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.