I'm looking for an example of elliptic curve which is isogenous but not isomorphism.
$[m]$ is typically map which is not isogeny which is not isomorphism.
For example, what is the image of multiplication by $2$ isogeny $[2]$ of $E:y^2＝x^3-x$ ?
Thank you in advance.
Best Answer
As noted by Jyrki in the comments, any non-constant morphism $C_1 \to C_2$ between smooth projective curves is surjective on the $\bar{K}$-points. In particular since for any $m \geq 1$ the multiplication by $m$ endomorphism $$[m] : E \to E$$ has image $E$. Since $m$ has degree $m^2$ such a morphism cannot be an isomorphism.
Such an isogeny is rather "trivial" in the sense that the kernel of $[m]$ is the whole of $E[m]$ which is not cyclic. Any (separable) isogeny between elliptic curves may be decomposed as $[m]$ and a cyclic isogeny (i.e., an isogeny with cyclic kernel).
However with your curve $E: y^2 = x^3 - x$ there is a rather well known $2$-isogeny between $E$ and $E': y^2 = x^3 + 4x$ given by $(x,y) \to (y^2/x^2, y(1-x^2)/x^2)$.
This is not an isomorphism since $(0,0)$ is a nontrivial element of its kernel.