An Example Over a Finite Field

Let $E$ be the elliptic curve $y^2 = x^3 + 3x + 3$ over the finite field

$$K=\mathbb{Z}/5\mathbb{Z}= \{0, 1, 2, 3, 4\}.$$

First, we find all points on $E$ using PARI:

? for(x=0,4, for(y=0,4, if((y^2-(x^3+3*x+3))%5==0, print1([x,y],"  "))))
[3, 2]  [3, 3]  [4, 2]  [4, 3]

Thus $E(K) = \left\{\O , (3,2), (3,3), (4,2), (4,3)\right\},$ so $E(K)$ must be a cyclic abelian group of order $5$. Let's verify that $E(K)$ is generated by $(3,2)$.

? e = ellinit([0,0,0,Mod(3,5),Mod(3,5)])
? ?ellpow    \\ type ?5 for a complete list of elliptic-curve functions
ellpow(e,x,n): n times the point x on elliptic curve e (n in Z).
? x = [3,2];
? for(n=1,5,print(n,"*[3,2] =  ",lift(ellpow(e,x,n))))
1*[3,2] =  [3, 2]
2*[3,2] =  [4, 3]
3*[3,2] =  [4, 2]
4*[3,2] =  [3, 3]
5*[3,2] =  [0]