Let $E_1$ and $E_2$ be elliptic curves defined over a field $\mathbb{K}$. An isogeny is a finite morphism $$\varphi:E_1\to E_2$$ such that $\varphi(\mathcal{O})=\mathcal{O}$. It is possible to show that an isogeny induces a morphisms of the groups of $E_1$ and $E_2$. It is clear that isogeny is an equivalence relation. Furthermore, following Hartshorne's Algebraic Geometry exercise 4.9.b, for a fixed elliptic curve $E$, the set elliptic curves of isogenous to $E$ is countable.
We are looking for a way to compute explicitly this set of isogenous curves for a fixed curve $E$. Our ultimate goal would be to give a set of curves $\lbrace E_i\rbrace_{i\in I}$ such that if $\varphi: E\to E'$ is an isogeny, then there exists $i\in I$ with $E_i\simeq E'$, each $E_i$ being expressed explicitely in the Weirstrass form (assuming $\mathrm{car}(\mathbb{K})\neq 2,3$).
Many theorems & propositions in Silverman's The Arithmetic of Elliptic Curves suggest a classification using subgroups of the fixed curve $E$. For example, proposition 4.12 tells us: "For an elliptic curve $E$ and $\Phi$ a finite subgroup of $E$, there is a unique elliptic curve $E'$ and a separable isogeny $\varphi:E\to E'$ such that $\ker \varphi=\Phi$." This approach did not show much progress yet unfortunatly.
I hope you find this question as interesting as I do. Any progress on the question, as tiny as it may seem, is more than welcome. If anything is unclear, please let me know.
Edit: Thanks to @Jyrki Lahtonen for pointing out this trivial case. Using Silverman's proposition cited above, if we have $\Phi=E[n]$, then the curve and the isogeny is $[n]:E\to E$.