Parametrization of line bundles over an elliptic curve by points of that curve

Question

Let $E$ be an elliptic curve over an algebraically closed field of characteristic zero, and let $\mathcal{L}$ be a line bundle on $E$ of degree $3$. Suppose, I can present this line bundle as $$ \mathcal{L} \cong \mathcal{O} (3p), $$ for some point $p \in E$. Is it true that point $p$ is unique? If not, how one can describe all points $p \in E$ with fixed class of the divisor $[3p]$?

Answer 1

$p$ is not unique, but any two possibilities for $p$ differ exactly by a 3-torsion point on $E$ (so there are exactly 9 possible values for $p$ that give the same divisor class). The basics of this are explained in chapter 3 of Silverman's AEC book, for example.

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The main tool here is Corollary III.3.5 in Silverman's Arithmetic of Elliptic Curves. It says that a divisor $D = \sum n_p \cdot (p)$ is principal if and only if it has degree zero and the sum (viewed as a sum using the group law on $E$) is equal to $O$, the marked point at infinity on $E$.

Clearly $3(p)\sim 3(p')$ if and only $3(p) - 3(p')$ is principal. Using the corollary mentioned above, this happens if and only if $3p \ominus 3p' = O$. But by commutativity of the group law, $3p \ominus 3p' = 3(p \ominus p')$, hence this happens if and only if $p$ and $p'$ differ by an element of $E[3]$.