I got the following question where I stuck at the moment. Given is the elliptic curve (EC) equation: $E: y^2+3xy+y=x^3+4x+4$ over the finite field ${\bf F}_5$ The first task is now to find out all possible points on the curve by just trying out all possible values for x.
For this I got the possible values: $E({\bf F}_5)=\{(2,0),(2,3),(4,1),P(inf)\}$ (don't know if its 100% correct) P(inf) is the infinite point of the curve.
The next task is to find the characteristic polynomial of the Frobenius endomorphism and to use this in order to calculate the number of points in $E({\bf F}_{25}).$
In order to get the characteristic polynomial I found for the trace of the Frobenius endomorphism is: $\alpha_5=5+1-t=4,$ so must be $t=2.$
To get now to the char. polyn. I have to find the roots for the equation of the char. polyn.: $X(T)=T^2-tT+q=T^2-2T+5.$
But my problem is now, that I cannot find any roots that fit to this equation $T^2+2T+5.$ Maybe you can help me? Thank you!
The roots of $T^2-2T+5$ are $\rho_{1,2}=1\pm 2{\rm i},$ so $$\#E({\bf F}_{p^2})=p^2+1-(\rho_1^2+\rho_2^2)=26-(-6)=32.$$