Determine groups for elliptic curves over a finite field

Question

I have the following question:

Determine all possible groups $E(F_5)$ for elliptic curves over $F_5$. What are their orders?

I am completely in the dark here about calculating the groups and their orders... I know how are the groups defined (how we calculate +, -...), I have Hasse's Theorem here somewhere, apparently an algorithm for calculating the order similar to "Baby Step, Giant Step" one, the Long-Trotter Method, Shank's Method, but I actually didn't know how to link them because I got no example in the course and I can not find anything on the internet.

Is calculating the number of elements giving me something about the group of the curve? I have listed all the possible curves regarding Weierstrass form and $27*b^2 + 4*a^3 \neq 0$, but I don't know how to proceed.

Answer 1

Using sage in a simpler manner we obtain the following detailed information:

sage: F = GF(5)
sage: curves = []
sage: for a, b in cartesian_product([F, F]):
....:     try:
....:         curves.append(EllipticCurve(F, [a, b]))
....:     except:
....:         print "Singular curve for a=%s b=%s" % (a, b)
....: 
Singular curve for a=0 b=0
Singular curve for a=2 b=2
Singular curve for a=2 b=3
Singular curve for a=3 b=1
Singular curve for a=3 b=4

sage: orders = list(set([ E.order() for E in curves ]))
sage: orders.sort()

sage: for ord in orders:
....:     print "ORDER", ord
....:     for E in curves:
....:         if E.order() != ord:    continue
....:         print '\t%s' % E
....:         print '\tGenerator(s):', ' AND '.join([ '%s of order %s' % (P.xy(), P.order()) for P in E.gens() ])
....:         print
....: 
ORDER 2
        Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 5
        Generator(s): (0, 0) of order 2

ORDER 3
        Elliptic Curve defined by y^2 = x^3 + 4*x + 2 over Finite Field of size 5
        Generator(s): (3, 1) of order 3

        Elliptic Curve defined by y^2 = x^3 + 4*x + 3 over Finite Field of size 5
        Generator(s): (2, 2) of order 3

ORDER 4
        Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 5
        Generator(s): (3, 0) of order 2 AND (2, 0) of order 2

        Elliptic Curve defined by y^2 = x^3 + x + 2 over Finite Field of size 5
        Generator(s): (1, 2) of order 4

        Elliptic Curve defined by y^2 = x^3 + x + 3 over Finite Field of size 5
        Generator(s): (4, 1) of order 4

ORDER 5
        Elliptic Curve defined by y^2 = x^3 + 3*x + 2 over Finite Field of size 5
        Generator(s): (1, 1) of order 5

        Elliptic Curve defined by y^2 = x^3 + 3*x + 3 over Finite Field of size 5
        Generator(s): (3, 2) of order 5

ORDER 6
        Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 5
        Generator(s): (2, 3) of order 6

        Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field of size 5
        Generator(s): (4, 4) of order 6

        Elliptic Curve defined by y^2 = x^3 + 3 over Finite Field of size 5
        Generator(s): (1, 3) of order 6

        Elliptic Curve defined by y^2 = x^3 + 4 over Finite Field of size 5
        Generator(s): (3, 1) of order 6

ORDER 7
        Elliptic Curve defined by y^2 = x^3 + 2*x + 1 over Finite Field of size 5
        Generator(s): (0, 1) of order 7

        Elliptic Curve defined by y^2 = x^3 + 2*x + 4 over Finite Field of size 5
        Generator(s): (4, 1) of order 7

ORDER 8
        Elliptic Curve defined by y^2 = x^3 + 4*x over Finite Field of size 5
        Generator(s): (2, 1) of order 4 AND (1, 0) of order 2

        Elliptic Curve defined by y^2 = x^3 + 4*x + 1 over Finite Field of size 5
        Generator(s): (0, 1) of order 8

        Elliptic Curve defined by y^2 = x^3 + 4*x + 4 over Finite Field of size 5
        Generator(s): (0, 2) of order 8

ORDER 9
        Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 5
        Generator(s): (4, 2) of order 9

        Elliptic Curve defined by y^2 = x^3 + x + 4 over Finite Field of size 5
        Generator(s): (2, 2) of order 9

ORDER 10
        Elliptic Curve defined by y^2 = x^3 + 3*x over Finite Field of size 5
        Generator(s): (3, 1) of order 10

sage: 

The Hasse bound insures a deviation of at most $2\sqrt 5$ for the number of $F$-rational points of an elliptic curve over $F=\Bbb F_5$, compared with the number of $F$-rational points ($5+1=6$) in the projective space $\Bbb P^1$ over $F$.

Possible values are thus $2,3,4,5,6,7,8,9,10$. All of them are taken for particular elliptic curves $E$.

In almost all cases we have a cyclic group $E(F)$ (and the generator has the order of the group of rational points of $E$ over $F$). In some special cases we have a $\Bbb Z/2\times \Bbb Z/d$ structure and two generators.