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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?

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  • $\begingroup$ As the answer says there is nothing to prove. $\endgroup$
    – Wuestenfux
    Commented Jun 27, 2019 at 10:57
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    $\begingroup$ Other examples over $\Bbb F_5$ can be found here. The list is not very difficult in the end. $\endgroup$ Commented Jun 27, 2019 at 12:41

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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.

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  • $\begingroup$ This was actually an example. I have also around 5 groups of order 6 and I don't know how to check if they are isomorphic $\endgroup$ Commented Jun 27, 2019 at 12:23
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    $\begingroup$ 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. $\endgroup$
    – Con
    Commented Jun 27, 2019 at 12:25
  • $\begingroup$ Just to make sure: That means all of them are isomorphic as groups as elliptic curves have the structure of abelian groups. $\endgroup$
    – Con
    Commented Jun 27, 2019 at 14:27

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