Artin representation 2.2e6_3e2_5e2.8t7.7c2

• Label: 2.2e6_3e2_5e2.8t7.7c2
• Dimension: 2
• Group: $C_8:C_2$
• Conductor: $ 2^{6} \cdot 3^{2} \cdot 5^{2}$
• Root number: not computed
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_8:C_2$
Conductor:	$14400= 2^{6} \cdot 3^{2} \cdot 5^{2} $
Artin number field:	Splitting field of $f= x^{8} - 120 x^{4} + 360 x^{2} + 720 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_8:C_2$
Parity:	Odd
Determinant:	1.5.4t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 151 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 7 + 85\cdot 151^{2} + 76\cdot 151^{3} + 139\cdot 151^{4} + 6\cdot 151^{5} + 43\cdot 151^{6} + 145\cdot 151^{7} +O\left(151^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 15 + 42\cdot 151 + 77\cdot 151^{2} + 68\cdot 151^{3} + 106\cdot 151^{4} + 132\cdot 151^{5} + 140\cdot 151^{6} + 121\cdot 151^{7} +O\left(151^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 22 + 26\cdot 151 + 132\cdot 151^{2} + 28\cdot 151^{3} + 91\cdot 151^{4} + 108\cdot 151^{5} + 13\cdot 151^{6} + 53\cdot 151^{7} +O\left(151^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 65 + 151 + 21\cdot 151^{2} + 21\cdot 151^{3} + 47\cdot 151^{4} + 88\cdot 151^{5} + 144\cdot 151^{6} + 110\cdot 151^{7} +O\left(151^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 86 + 149\cdot 151 + 129\cdot 151^{2} + 129\cdot 151^{3} + 103\cdot 151^{4} + 62\cdot 151^{5} + 6\cdot 151^{6} + 40\cdot 151^{7} +O\left(151^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 129 + 124\cdot 151 + 18\cdot 151^{2} + 122\cdot 151^{3} + 59\cdot 151^{4} + 42\cdot 151^{5} + 137\cdot 151^{6} + 97\cdot 151^{7} +O\left(151^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 136 + 108\cdot 151 + 73\cdot 151^{2} + 82\cdot 151^{3} + 44\cdot 151^{4} + 18\cdot 151^{5} + 10\cdot 151^{6} + 29\cdot 151^{7} +O\left(151^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 144 + 150\cdot 151 + 65\cdot 151^{2} + 74\cdot 151^{3} + 11\cdot 151^{4} + 144\cdot 151^{5} + 107\cdot 151^{6} + 5\cdot 151^{7} +O\left(151^{ 8 }\right)$

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

Cycle notation

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

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

$(3,6)(4,5)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.