Properties

 Label 2.1280.8t17.a Dimension $2$ Group $C_4\wr C_2$ Conductor $1280$ Indicator $0$

Related objects

Basic invariants

 Dimension: $2$ Group: $C_4\wr C_2$ Conductor: $$1280$$$$\medspace = 2^{8} \cdot 5$$ Artin number field: Galois closure of 8.0.2097152000.3 Galois orbit size: $2$ Smallest permutation container: $C_4\wr C_2$ Parity: odd Projective image: $D_4$ Projective field: Galois closure of 4.2.2000.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 109 }$ to precision 7.
Roots:
 $r_{ 1 }$ $=$ $$11 + 11\cdot 109 + 77\cdot 109^{3} + 19\cdot 109^{4} + 18\cdot 109^{5} + 84\cdot 109^{6} +O(109^{7})$$ 11 + 11*109 + 77*109^3 + 19*109^4 + 18*109^5 + 84*109^6+O(109^7) $r_{ 2 }$ $=$ $$25 + 108\cdot 109 + 92\cdot 109^{2} + 2\cdot 109^{3} + 41\cdot 109^{4} + 21\cdot 109^{5} + 48\cdot 109^{6} +O(109^{7})$$ 25 + 108*109 + 92*109^2 + 2*109^3 + 41*109^4 + 21*109^5 + 48*109^6+O(109^7) $r_{ 3 }$ $=$ $$36 + 109 + 24\cdot 109^{2} + 77\cdot 109^{3} + 58\cdot 109^{4} + 6\cdot 109^{5} + 75\cdot 109^{6} +O(109^{7})$$ 36 + 109 + 24*109^2 + 77*109^3 + 58*109^4 + 6*109^5 + 75*109^6+O(109^7) $r_{ 4 }$ $=$ $$47 + 42\cdot 109 + 74\cdot 109^{2} + 39\cdot 109^{3} + 89\cdot 109^{4} + 74\cdot 109^{5} + 53\cdot 109^{6} +O(109^{7})$$ 47 + 42*109 + 74*109^2 + 39*109^3 + 89*109^4 + 74*109^5 + 53*109^6+O(109^7) $r_{ 5 }$ $=$ $$62 + 66\cdot 109 + 34\cdot 109^{2} + 69\cdot 109^{3} + 19\cdot 109^{4} + 34\cdot 109^{5} + 55\cdot 109^{6} +O(109^{7})$$ 62 + 66*109 + 34*109^2 + 69*109^3 + 19*109^4 + 34*109^5 + 55*109^6+O(109^7) $r_{ 6 }$ $=$ $$73 + 107\cdot 109 + 84\cdot 109^{2} + 31\cdot 109^{3} + 50\cdot 109^{4} + 102\cdot 109^{5} + 33\cdot 109^{6} +O(109^{7})$$ 73 + 107*109 + 84*109^2 + 31*109^3 + 50*109^4 + 102*109^5 + 33*109^6+O(109^7) $r_{ 7 }$ $=$ $$84 + 16\cdot 109^{2} + 106\cdot 109^{3} + 67\cdot 109^{4} + 87\cdot 109^{5} + 60\cdot 109^{6} +O(109^{7})$$ 84 + 16*109^2 + 106*109^3 + 67*109^4 + 87*109^5 + 60*109^6+O(109^7) $r_{ 8 }$ $=$ $$98 + 97\cdot 109 + 108\cdot 109^{2} + 31\cdot 109^{3} + 89\cdot 109^{4} + 90\cdot 109^{5} + 24\cdot 109^{6} +O(109^{7})$$ 98 + 97*109 + 108*109^2 + 31*109^3 + 89*109^4 + 90*109^5 + 24*109^6+O(109^7)

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

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

Character values on conjugacy classes

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