What is the kernel polynomial of a degree $l$ isogeny? Given an elliptic curve $E$ in Weierstrass form. Is there a  standard way to describe the kernel of a degree $l$ separable isogeny? More specifically, can the kernel polynomial be written down, only knowing $E$ and that the isogeny is separable and has degree $l$?
My motivation is that given a a curve $E$ that parametrizes curves with degree $l$ isogenies, I want to be able to explicitly figure out what the isogeny and co-domain is. I have read that Velu's and Kohel's methods can be used if you have the curve equation $E$ and the kernel as either a set of points, or the kernel polynomial.
 A: I'm not sure what you mean by a standard way, here's an answer from someone working with elliptic curves related to cryptography. 
Steven Galbraith has a nice chapter on Elliptic Curves in his book Mathematics of Public Key Cryptography. Theorem 9.6.19 describes the correspondence you indeed mention, i.e. up to isomorphism we can think of separable $l$-isogenies as subgroups of $E(k)$ of order $l$. Theorem 9.7.5 explains that we can represent the isogeny with some rational function $\phi_1(x)$ which we can compute using Velu's formulas. However, in crypto we are usually working with subgroups of very large order, which as a consequence makes the degree of $\phi_1$ very large. Therefore it can be infeasible to represent $\phi$ this way.
A relatively new post-quantum scheme called supersingular-isogeny-based cryptography actually uses some $l$-isogeny as a secret key, where $l=p^n$ for a small prime $p$ (so far $l=2,3$ have been proposed) and $n$ suitably chosen to obtain a high enough security level (think of $n$ somewhere in $[200,1000]$). This is an exponentially large degree, hence the isogeny cannot be represented with the polynomial. Instead, one only considers isogenies with cyclic kernel. Therefore we can represent an isogeny by a single point $P\in E[l]$. Even more, using the fact that $E[l]\cong \mathbb{Z}/l\mathbb{Z}\times\mathbb{Z}/l\mathbb{Z}$, we can represent $\phi$ by two elements $a,b\in\mathbb{Z}/l\mathbb{Z}$.
This is probably a very one-sided perspective, but I believe it showcases that the best way to represent an isogeny may very much depend on context.
