TITLE: Veronese embeddings of elliptic curves in weighted projective space
QUESTION [5 upvotes]: 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 (if this is even true)? For example, $C_7\subset\mathbb{P}(1,2,3)$.

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.

REPLY [3 votes]: You are just considering the map defined by a linear system $|D_k|$ of degree $k$ (that $D_k=kp$ is irrelevant, any degree $k$ divisor on $E$ is linearly equivalent to $kp$ for some $p$). Any introductory book on curves will tell you that this is an embedding in $\mathbb{P}^{k-1}$ for $k\geq 3$.
This gives only one embedding type for each $k$, so other types do not appear in this way.