# Properties

 Label 2.1280.8t17.d.b Dimension $2$ Group $C_4\wr C_2$ Conductor $1280$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_4\wr C_2$ Conductor: $$1280$$$$\medspace = 2^{8} \cdot 5$$ Artin stem field: Galois closure of 8.0.2097152000.8 Galois orbit size: $2$ Smallest permutation container: $C_4\wr C_2$ Parity: odd Determinant: 1.5.4t1.a.b Projective image: $D_4$ Projective stem field: Galois closure of 4.2.8000.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} + 4x^{4} + 20$$ x^8 + 4*x^4 + 20 .

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 9.

Roots:
 $r_{ 1 }$ $=$ $$5 + 31\cdot 61 + 54\cdot 61^{2} + 4\cdot 61^{3} + 49\cdot 61^{4} + 54\cdot 61^{6} + 55\cdot 61^{7} + 25\cdot 61^{8} +O(61^{9})$$ 5 + 31*61 + 54*61^2 + 4*61^3 + 49*61^4 + 54*61^6 + 55*61^7 + 25*61^8+O(61^9) $r_{ 2 }$ $=$ $$6 + 30\cdot 61 + 43\cdot 61^{2} + 7\cdot 61^{3} + 42\cdot 61^{4} + 6\cdot 61^{5} + 2\cdot 61^{6} + 25\cdot 61^{7} + 24\cdot 61^{8} +O(61^{9})$$ 6 + 30*61 + 43*61^2 + 7*61^3 + 42*61^4 + 6*61^5 + 2*61^6 + 25*61^7 + 24*61^8+O(61^9) $r_{ 3 }$ $=$ $$25 + 25\cdot 61 + 48\cdot 61^{2} + 34\cdot 61^{3} + 29\cdot 61^{4} + 24\cdot 61^{5} + 43\cdot 61^{6} + 29\cdot 61^{7} + 40\cdot 61^{8} +O(61^{9})$$ 25 + 25*61 + 48*61^2 + 34*61^3 + 29*61^4 + 24*61^5 + 43*61^6 + 29*61^7 + 40*61^8+O(61^9) $r_{ 4 }$ $=$ $$30 + 55\cdot 61 + 25\cdot 61^{2} + 21\cdot 61^{3} + 45\cdot 61^{4} + 20\cdot 61^{5} + 16\cdot 61^{6} + 57\cdot 61^{7} + 60\cdot 61^{8} +O(61^{9})$$ 30 + 55*61 + 25*61^2 + 21*61^3 + 45*61^4 + 20*61^5 + 16*61^6 + 57*61^7 + 60*61^8+O(61^9) $r_{ 5 }$ $=$ $$31 + 5\cdot 61 + 35\cdot 61^{2} + 39\cdot 61^{3} + 15\cdot 61^{4} + 40\cdot 61^{5} + 44\cdot 61^{6} + 3\cdot 61^{7} +O(61^{9})$$ 31 + 5*61 + 35*61^2 + 39*61^3 + 15*61^4 + 40*61^5 + 44*61^6 + 3*61^7+O(61^9) $r_{ 6 }$ $=$ $$36 + 35\cdot 61 + 12\cdot 61^{2} + 26\cdot 61^{3} + 31\cdot 61^{4} + 36\cdot 61^{5} + 17\cdot 61^{6} + 31\cdot 61^{7} + 20\cdot 61^{8} +O(61^{9})$$ 36 + 35*61 + 12*61^2 + 26*61^3 + 31*61^4 + 36*61^5 + 17*61^6 + 31*61^7 + 20*61^8+O(61^9) $r_{ 7 }$ $=$ $$55 + 30\cdot 61 + 17\cdot 61^{2} + 53\cdot 61^{3} + 18\cdot 61^{4} + 54\cdot 61^{5} + 58\cdot 61^{6} + 35\cdot 61^{7} + 36\cdot 61^{8} +O(61^{9})$$ 55 + 30*61 + 17*61^2 + 53*61^3 + 18*61^4 + 54*61^5 + 58*61^6 + 35*61^7 + 36*61^8+O(61^9) $r_{ 8 }$ $=$ $$56 + 29\cdot 61 + 6\cdot 61^{2} + 56\cdot 61^{3} + 11\cdot 61^{4} + 60\cdot 61^{5} + 6\cdot 61^{6} + 5\cdot 61^{7} + 35\cdot 61^{8} +O(61^{9})$$ 56 + 29*61 + 6*61^2 + 56*61^3 + 11*61^4 + 60*61^5 + 6*61^6 + 5*61^7 + 35*61^8+O(61^9)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.