Moduli space of isogeny classes of elliptic curves The modular curve $Y(1)$ classifies isomorphism classes of elliptic curves, namely its $K$-points for any field $\mathbb Q\subseteq K\subseteq \mathbb C$ correspond via the $j$-invariant to $\mathbb C$-isomorphism classes of elliptic curves defined over $K$.
My question is: what if one wants to classify isogeny classes of elliptic curves defined over $K$? Is there an appropriate moduli space for that? Namely, is there an algebraic variety $Y$ whose $K$-points correspond functorially to $\mathbb C$-isogeny classes (or even better $K$-isogeny classes) of elliptic curves defined over $K$?

When we try to classify elliptic curves up to isogeny, then we use the modular curves $X_0(N)/\mathbb{Q}$, whose non-cuspidal $K$-rational points (for some number field $K$) classify triples $(E/K,E'/K,\phi/K)$ of elliptic curves $E$ and $E'$ defined over $K$, together with an isogeny $\phi:E\to E'$ defined over $K$, with cyclic kernel of size $N$. In other words, the non-cuspidal points on $X_0(N)$ classify $N$-isogenies of elliptic curves.
For instance, $X_0(11)$ has three non-cuspidal $\mathbb{Q}$-rational points, that correspond to the only three $11$-isogenies $121A1\to 121A2$, and $121C1\to 121C2$, and $121B1\to 121B2$, where here I am using Cremona's notation to label elliptic curves. The curve $X_0(11)$ by the way is itself elliptic, it is the curve $11A1$ with a model
$$y^2+y=x^3-x^2-10x-20.$$
The Mordell-Weil group of $X_0(11)(\mathbb{Q})$ is isomorphic to $\mathbb{Z}/5\mathbb{Z}$, generated by the point $(16,60)$. Two of the $5$-torsion points correspond to cusps, while the other three correspond to the isogenies mentioned above.
