# Properties

 Label 2.160.8t17.a.b Dimension $2$ Group $C_4\wr C_2$ Conductor $160$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_4\wr C_2$ Conductor: $$160$$$$\medspace = 2^{5} \cdot 5$$ Artin stem field: Galois closure of 8.0.8192000.1 Galois orbit size: $2$ Smallest permutation container: $C_4\wr C_2$ Parity: odd Determinant: 1.5.4t1.a.a Projective image: $D_4$ Projective stem field: Galois closure of 4.2.2000.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 2x^{4} + 5$$ x^8 - 2*x^4 + 5 .

The roots of $f$ are computed in $\Q_{ 181 }$ to precision 6.

Roots:
 $r_{ 1 }$ $=$ $$13 + 27\cdot 181 + 151\cdot 181^{2} + 165\cdot 181^{3} + 106\cdot 181^{4} + 35\cdot 181^{5} +O(181^{6})$$ 13 + 27*181 + 151*181^2 + 165*181^3 + 106*181^4 + 35*181^5+O(181^6) $r_{ 2 }$ $=$ $$62 + 22\cdot 181 + 172\cdot 181^{2} + 143\cdot 181^{3} + 44\cdot 181^{4} + 40\cdot 181^{5} +O(181^{6})$$ 62 + 22*181 + 172*181^2 + 143*181^3 + 44*181^4 + 40*181^5+O(181^6) $r_{ 3 }$ $=$ $$66 + 37\cdot 181 + 46\cdot 181^{2} + 81\cdot 181^{3} + 159\cdot 181^{4} + 52\cdot 181^{5} +O(181^{6})$$ 66 + 37*181 + 46*181^2 + 81*181^3 + 159*181^4 + 52*181^5+O(181^6) $r_{ 4 }$ $=$ $$89 + 26\cdot 181 + 149\cdot 181^{2} + 5\cdot 181^{3} + 90\cdot 181^{4} + 85\cdot 181^{5} +O(181^{6})$$ 89 + 26*181 + 149*181^2 + 5*181^3 + 90*181^4 + 85*181^5+O(181^6) $r_{ 5 }$ $=$ $$92 + 154\cdot 181 + 31\cdot 181^{2} + 175\cdot 181^{3} + 90\cdot 181^{4} + 95\cdot 181^{5} +O(181^{6})$$ 92 + 154*181 + 31*181^2 + 175*181^3 + 90*181^4 + 95*181^5+O(181^6) $r_{ 6 }$ $=$ $$115 + 143\cdot 181 + 134\cdot 181^{2} + 99\cdot 181^{3} + 21\cdot 181^{4} + 128\cdot 181^{5} +O(181^{6})$$ 115 + 143*181 + 134*181^2 + 99*181^3 + 21*181^4 + 128*181^5+O(181^6) $r_{ 7 }$ $=$ $$119 + 158\cdot 181 + 8\cdot 181^{2} + 37\cdot 181^{3} + 136\cdot 181^{4} + 140\cdot 181^{5} +O(181^{6})$$ 119 + 158*181 + 8*181^2 + 37*181^3 + 136*181^4 + 140*181^5+O(181^6) $r_{ 8 }$ $=$ $$168 + 153\cdot 181 + 29\cdot 181^{2} + 15\cdot 181^{3} + 74\cdot 181^{4} + 145\cdot 181^{5} +O(181^{6})$$ 168 + 153*181 + 29*181^2 + 15*181^3 + 74*181^4 + 145*181^5+O(181^6)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.