Galois orbit of Artin representations 2.3_5_11.8t17.2

• Label: 2.3_5_11.8t17.2
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 3 \cdot 5 \cdot 11 $
• 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} + 5 x^{6} - 2 x^{5} + 9 x^{4} - 2 x^{3} + 5 x^{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 $\Q_{ 59 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 2 + 24\cdot 59 + 48\cdot 59^{2} + 5\cdot 59^{3} + 40\cdot 59^{4} + 32\cdot 59^{5} +O\left(59^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 16 + 35\cdot 59 + 21\cdot 59^{2} + 42\cdot 59^{3} + 18\cdot 59^{4} + 27\cdot 59^{5} +O\left(59^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 19 + 44\cdot 59 + 57\cdot 59^{2} + 7\cdot 59^{3} + 11\cdot 59^{4} + 37\cdot 59^{5} +O\left(59^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 28 + 3\cdot 59 + 29\cdot 59^{2} + 45\cdot 59^{3} + 19\cdot 59^{4} + 11\cdot 59^{5} +O\left(59^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 30 + 23\cdot 59 + 30\cdot 59^{2} + 28\cdot 59^{3} + 39\cdot 59^{4} + 57\cdot 59^{5} +O\left(59^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 42 + 36\cdot 59 + 24\cdot 59^{2} + 48\cdot 59^{3} + 11\cdot 59^{4} + 13\cdot 59^{5} +O\left(59^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 48 + 38\cdot 59 + 10\cdot 59^{2} + 54\cdot 59^{3} + 2\cdot 59^{4} + 39\cdot 59^{5} +O\left(59^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 52 + 29\cdot 59 + 13\cdot 59^{2} + 3\cdot 59^{3} + 33\cdot 59^{4} + 17\cdot 59^{5} +O\left(59^{ 6 }\right)$

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

Cycle notation

$(2,3,7,4)$

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

$(2,7)(3,4)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.