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If we have two elliptic curves $E$ and $E'$ and the points of both elliptic curves are same. Then all the points of $E$ map to all the points of another elliptic curve $E'$.

For example $E$ has seven points and $E`$ also have seven points then would isogeny make this possible?

Is the reverse $r$ possible by dual isogeny mapping? In other words, could all the points of $E'$ map to all the points of $E$?

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    $\begingroup$ Welcome to crypto.stackexchange - here is some advice to help you receive better reception and answers to your question: What you are asking is unclear. Please try to re-formulate your question. Grammar improvements as well as removing the marked text would help. You can use the "edit" link below the question and tags to make changes. $\endgroup$
    – Ella Rose
    Mar 13, 2019 at 22:39
  • $\begingroup$ I've tried my best to fix the question, but it is likely still unclear. I've translated "Saveen" to "seven", presuming that's what was meant. Please reread your question or show it to a friend before posting. $\endgroup$
    – Maarten Bodewes ♦
    Mar 13, 2019 at 22:53

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I'm not sure what is being asked, but here are some related facts. If $E$ and $E'$ are elliptic curves over a finite field $K$, then there exists an isogeny $\Phi:E\to E'$ defined over $K$ if and only if the groups $E(K)$ and $E'(K)$ have the same cardinality. However, I note that there are examples where $E(K)$ and $E'(K)$ have the same cardinality but are not isomorphic. In such a situation, every choice of the isogeny $\Phi$ would have kernel containing some nonidentity point in $E(K)$, and also each such isogeny would induce a function $E(K)\to E'(K)$ which is not surjective.

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