Artin representation 2.2e6_3e2_5e2.8t7.6c1

• Label: 2.2e6_3e2_5e2.8t7.6c1
• 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:	1.5.4t1.1c2

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 181 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 32 + 133\cdot 181 + 121\cdot 181^{2} + 88\cdot 181^{3} + 115\cdot 181^{4} + 42\cdot 181^{5} + 143\cdot 181^{6} +O\left(181^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 62 + 69\cdot 181 + 101\cdot 181^{2} + 17\cdot 181^{3} + 123\cdot 181^{4} + 45\cdot 181^{5} + 173\cdot 181^{6} +O\left(181^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 67 + 113\cdot 181 + 139\cdot 181^{2} + 57\cdot 181^{3} + 180\cdot 181^{4} + 125\cdot 181^{5} + 103\cdot 181^{6} +O\left(181^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 69 + 170\cdot 181 + 134\cdot 181^{2} + 58\cdot 181^{3} + 178\cdot 181^{4} + 76\cdot 181^{5} + 137\cdot 181^{6} +O\left(181^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 112 + 10\cdot 181 + 46\cdot 181^{2} + 122\cdot 181^{3} + 2\cdot 181^{4} + 104\cdot 181^{5} + 43\cdot 181^{6} +O\left(181^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 114 + 67\cdot 181 + 41\cdot 181^{2} + 123\cdot 181^{3} + 55\cdot 181^{5} + 77\cdot 181^{6} +O\left(181^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 119 + 111\cdot 181 + 79\cdot 181^{2} + 163\cdot 181^{3} + 57\cdot 181^{4} + 135\cdot 181^{5} + 7\cdot 181^{6} +O\left(181^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 149 + 47\cdot 181 + 59\cdot 181^{2} + 92\cdot 181^{3} + 65\cdot 181^{4} + 138\cdot 181^{5} + 37\cdot 181^{6} +O\left(181^{ 7 }\right)$

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

Cycle notation

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

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

$(1,8)(2,7)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.