# Properties

 Label 2.1280.8t6.d Dimension $2$ Group $D_{8}$ Conductor $1280$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{8}$ Conductor: $$1280$$$$\medspace = 2^{8} \cdot 5$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 8.0.2097152000.6 Galois orbit size: $2$ Smallest permutation container: $D_{8}$ Parity: odd Projective image: $D_4$ Projective field: Galois closure of 4.0.1280.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 109 }$ to precision 6.
Roots:
 $r_{ 1 }$ $=$ $$18 + 45\cdot 109 + 81\cdot 109^{2} + 92\cdot 109^{3} + 27\cdot 109^{5} +O(109^{6})$$ 18 + 45*109 + 81*109^2 + 92*109^3 + 27*109^5+O(109^6) $r_{ 2 }$ $=$ $$22 + 10\cdot 109 + 12\cdot 109^{2} + 9\cdot 109^{3} + 70\cdot 109^{4} + 50\cdot 109^{5} +O(109^{6})$$ 22 + 10*109 + 12*109^2 + 9*109^3 + 70*109^4 + 50*109^5+O(109^6) $r_{ 3 }$ $=$ $$26 + 99\cdot 109 + 5\cdot 109^{2} + 99\cdot 109^{3} + 12\cdot 109^{4} + 102\cdot 109^{5} +O(109^{6})$$ 26 + 99*109 + 5*109^2 + 99*109^3 + 12*109^4 + 102*109^5+O(109^6) $r_{ 4 }$ $=$ $$41 + 46\cdot 109 + 7\cdot 109^{2} + 56\cdot 109^{3} + 84\cdot 109^{4} + 79\cdot 109^{5} +O(109^{6})$$ 41 + 46*109 + 7*109^2 + 56*109^3 + 84*109^4 + 79*109^5+O(109^6) $r_{ 5 }$ $=$ $$68 + 62\cdot 109 + 101\cdot 109^{2} + 52\cdot 109^{3} + 24\cdot 109^{4} + 29\cdot 109^{5} +O(109^{6})$$ 68 + 62*109 + 101*109^2 + 52*109^3 + 24*109^4 + 29*109^5+O(109^6) $r_{ 6 }$ $=$ $$83 + 9\cdot 109 + 103\cdot 109^{2} + 9\cdot 109^{3} + 96\cdot 109^{4} + 6\cdot 109^{5} +O(109^{6})$$ 83 + 9*109 + 103*109^2 + 9*109^3 + 96*109^4 + 6*109^5+O(109^6) $r_{ 7 }$ $=$ $$87 + 98\cdot 109 + 96\cdot 109^{2} + 99\cdot 109^{3} + 38\cdot 109^{4} + 58\cdot 109^{5} +O(109^{6})$$ 87 + 98*109 + 96*109^2 + 99*109^3 + 38*109^4 + 58*109^5+O(109^6) $r_{ 8 }$ $=$ $$91 + 63\cdot 109 + 27\cdot 109^{2} + 16\cdot 109^{3} + 108\cdot 109^{4} + 81\cdot 109^{5} +O(109^{6})$$ 91 + 63*109 + 27*109^2 + 16*109^3 + 108*109^4 + 81*109^5+O(109^6)

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $2$ $2$ $1$ $2$ $(1,8)(2,7)(3,6)(4,5)$ $-2$ $-2$ $4$ $2$ $(1,3)(2,4)(5,7)(6,8)$ $0$ $0$ $4$ $2$ $(1,7)(2,8)(3,6)$ $0$ $0$ $2$ $4$ $(1,2,8,7)(3,5,6,4)$ $0$ $0$ $2$ $8$ $(1,6,2,4,8,3,7,5)$ $-\zeta_{8}^{3} + \zeta_{8}$ $\zeta_{8}^{3} - \zeta_{8}$ $2$ $8$ $(1,4,7,6,8,5,2,3)$ $\zeta_{8}^{3} - \zeta_{8}$ $-\zeta_{8}^{3} + \zeta_{8}$
The blue line marks the conjugacy class containing complex conjugation.