# Prove that two groups for elliptic curves are isomorphic

I was asked to calculate all the possible groups for elliptic curves and their order in $$\mathbb{F}_5$$. There are $$p^2-p$$ groups that respect $$\Delta \neq 0$$, so there are $$20$$ groups.

Some of them may be isomorphic. I have to look for the ones with the same order.

For example: are the ones defined by $$x^3+4x+2$$ and $$x^3+4x+3$$ isomorphic? The points of the first groups are $$(3,1), (3,4), (\infty, \infty)$$, for the second group are $$(2,2), (2,3), (\infty, \infty)$$

How do I check if they are isomorphic or not?

• As the answer says there is nothing to prove. – Wuestenfux Jun 27 at 10:57
• Other examples over $\Bbb F_5$ can be found here. The list is not very difficult in the end. – Dietrich Burde Jun 27 at 12:41

If I understand you correctly you have two groups of order three which means they are isomorphic as there only is one group of order $$3$$ up to isomorphism.
• Groups of order 6 are either cyclic (therefore also abelian) or isomorphic to $S_3$. So you could check whether they are abelian or not. – ThorWittich Jun 27 at 12:25