Artin representation 2.3_5_11.8t17.3c2

• Label: 2.3_5_11.8t17.3c2
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 3 \cdot 5 \cdot 11 $
• Root number: not computed
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$165= 3 \cdot 5 \cdot 11 $
Artin number field:	Splitting field of $f= x^{8} - x^{7} + x^{6} + 2 x^{5} - 3 x^{4} - 2 x^{3} + x^{2} + x + 1 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_4\wr C_2$
Parity:	Odd
Determinant:	1.3_5_11.4t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 269 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 52 + 39\cdot 269 + 148\cdot 269^{2} + 164\cdot 269^{3} + 60\cdot 269^{4} +O\left(269^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 60 + 178\cdot 269 + 199\cdot 269^{2} + 152\cdot 269^{3} + 180\cdot 269^{4} +O\left(269^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 70 + 90\cdot 269 + 188\cdot 269^{2} + 123\cdot 269^{3} + 265\cdot 269^{4} +O\left(269^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 73 + 25\cdot 269 + 98\cdot 269^{2} + 200\cdot 269^{3} + 195\cdot 269^{4} +O\left(269^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 128 + 88\cdot 269 + 56\cdot 269^{2} + 41\cdot 269^{3} + 179\cdot 269^{4} +O\left(269^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 130 + 246\cdot 269 + 231\cdot 269^{2} + 258\cdot 269^{3} + 263\cdot 269^{4} +O\left(269^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 145 + 70\cdot 269 + 16\cdot 269^{2} + 109\cdot 269^{3} + 186\cdot 269^{4} +O\left(269^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 150 + 68\cdot 269 + 137\cdot 269^{2} + 25\cdot 269^{3} + 13\cdot 269^{4} +O\left(269^{ 5 }\right)$

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

Cycle notation

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

$(2,4,6,3)$

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

$(2,6)(3,4)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.