For each of the $15$ possible torsion groups of an elliptic curve defined over $\mathbb{Q}$ we have an infinite family of curves with that torsion group. This sometimes goes under the name of Kubert normal form or Tate normal form.

I have been wondering if we have something similar for the following setting.

Let's say we have an elliptic curve $E$ with torsion group $T$ and an elliptic curve $E'$ with torsion group $T'$ and an isogeny $E \rightarrow E'$.

Is it possible to come up with infinite families of such pairs of isogenous curves $E, E'$ for each (or some) of the $15 \times 14$ pairs of torsion groups $T, T'$?

Or are there any other partial results related to this question?


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