Let $E$ be an elliptic curve and $D_k=kp$ a divisor on $E$, where $p\in E$, for $k\in\mathbb{N}$. Then we can reconstruct $E$ from the graded ring $R(D_k)=\bigoplus_{n\geqslant0}\mathcal{L}({nD_k})$: $$E\cong \mathrm{Proj}\bigoplus_{n\geqslant0}\mathcal{L}(nkp).$$

Below is a table of what this results in for the first few values of $k$.

(A really useful resource here is this problem sheet by Miles Reid.)

I have two questions for which I can't seem to find any reference:

- What happens if we carry on -- can we get an embedding for every value of $k\in\mathbb{N}$? If so, are they all different enough to be interesting, or is there a reason that most sources seem to stop after $5$?
- Why do some embeddings never occur in the above sequence
? For example, $C_7\subset\mathbb{P}(1,2,3)$.*(if this is even true)*

**Edit:**
The answer to 1. is apparently that we can keep on going, obtaining embeddings in $\mathbb{P}^{k-1}$ (since $D_k$ is very ample for $k\geqslant3$), but they do get increasingly complicated.