Galois orbit of Artin representations 2.3e2_61e2.8t17.1

• Label: 2.3e2_61e2.8t17.1
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 3^{2} \cdot 61^{2}$
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$33489= 3^{2} \cdot 61^{2} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} + 2 x^{6} - 6 x^{5} + 30 x^{4} - 63 x^{3} + 71 x^{2} - 36 x + 7 $ 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 $\Q_{ 73 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 21 + 27\cdot 73 + 45\cdot 73^{2} + 14\cdot 73^{3} + 2\cdot 73^{4} + 51\cdot 73^{5} +O\left(73^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 23 + 17\cdot 73 + 21\cdot 73^{2} + 65\cdot 73^{3} + 59\cdot 73^{5} +O\left(73^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 40 + 16\cdot 73 + 17\cdot 73^{2} + 20\cdot 73^{3} + 47\cdot 73^{5} +O\left(73^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 42 + 56\cdot 73 + 54\cdot 73^{2} + 61\cdot 73^{3} + 48\cdot 73^{4} + 16\cdot 73^{5} +O\left(73^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 51 + 12\cdot 73 + 50\cdot 73^{2} + 61\cdot 73^{3} + 60\cdot 73^{4} + 34\cdot 73^{5} +O\left(73^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 58 + 69\cdot 73 + 55\cdot 73^{2} + 62\cdot 73^{3} + 70\cdot 73^{4} + 24\cdot 73^{5} +O\left(73^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 61 + 49\cdot 73 + 3\cdot 73^{2} + 48\cdot 73^{3} + 61\cdot 73^{4} + 12\cdot 73^{5} +O\left(73^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 72 + 41\cdot 73 + 43\cdot 73^{2} + 30\cdot 73^{3} + 46\cdot 73^{4} + 45\cdot 73^{5} +O\left(73^{ 6 }\right)$

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

Cycle notation

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

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

$(4,7)(5,6)$

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

$(4,6,7,5)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.