Let $\phi\!: C_1 \to C_2$ be a separable morphism of smooth irreducible curves embedded as projectively normal models by invertible sheaves $\mathcal{L}_1$ and $\mathcal{L}_2$ respectively. Theorem 4.1 of Kohel's article (cf. Lange's article) claims that the morphism $\phi$ can be given by a complete tuple $\mathfrak{s} = (f_0, \cdots\!, f_r)$ of defining polynomials of degree $n \in \mathbb{N}$ if (and only if) $\phi^*\mathcal{L}_2 \simeq \mathcal{L}_1^n$. In the proof of the theorem it is said that the tuple $\mathfrak{s}$ corresponds to an element of the one-dimensional vector space $$ \mathrm{Hom}(\phi^*\mathcal{L}_2, \mathcal{L}_1^n) \simeq \Gamma(C_1, \phi^*\mathcal{L}_2^{-1} \otimes \mathcal{L}_1^n). $$ Suppose that I know a non-zero rational function from the right space. How to construct explicitly the corresponding tuple $\mathfrak{s}$ ?
In fact, I am interested in the case when $\phi$ is a cyclic isogeny of odd degree (with the given kernel $G$) between elliptic curves $C_1$, $C_2$ in the Weierstrass form, $n=1$, and $\mathcal{L}_1$, $\mathcal{L}_2$ correspond to the divisors $3\deg(\phi)\mathcal{O}$ and $3\mathcal{O}$ respectively, where $\mathcal{O} := (0:1:0)$. It is easily seen that the condition $\phi^*\mathcal{L}_2 \simeq \mathcal{L}_1^n$ is met, because $\#G = \deg(\phi)$ and the sum of points from $G$ is zero.
