Galois orbit of Artin representations 2.2e2_5e2_13e2.8t17.2

• Label: 2.2e2_5e2_13e2.8t17.2
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 2^{2} \cdot 5^{2} \cdot 13^{2}$
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$16900= 2^{2} \cdot 5^{2} \cdot 13^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} - 2 x^{6} + 11 x^{4} - 2 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 $\Q_{ 73 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 10 + 34\cdot 73 + 23\cdot 73^{2} + 42\cdot 73^{3} + 28\cdot 73^{4} + 45\cdot 73^{5} +O\left(73^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 22 + 49\cdot 73 + 22\cdot 73^{2} + 38\cdot 73^{3} + 73^{4} + 27\cdot 73^{5} +O\left(73^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 23 + 64\cdot 73 + 25\cdot 73^{2} + 73^{3} + 70\cdot 73^{4} + 56\cdot 73^{5} +O\left(73^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 28 + 7\cdot 73 + 65\cdot 73^{2} + 45\cdot 73^{3} + 48\cdot 73^{4} + 58\cdot 73^{5} +O\left(73^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 29 + 36\cdot 73 + 55\cdot 73^{2} + 47\cdot 73^{3} + 57\cdot 73^{4} + 19\cdot 73^{5} +O\left(73^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 54 + 68\cdot 73 + 3\cdot 73^{2} + 23\cdot 73^{3} + 21\cdot 73^{4} + 22\cdot 73^{5} +O\left(73^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 60 + 65\cdot 73 + 5\cdot 73^{2} + 36\cdot 73^{3} + 8\cdot 73^{4} + 35\cdot 73^{5} +O\left(73^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 68 + 38\cdot 73 + 16\cdot 73^{2} + 57\cdot 73^{3} + 55\cdot 73^{4} + 26\cdot 73^{5} +O\left(73^{ 6 }\right)$

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

Cycle notation

$(1,2)(4,7)$

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

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

$(1,7,2,4)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.