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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?

Jesper Petersen
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    Clearly I am late. If you want a cyclic torsion subgroup of order $n$ and an isogeny of degree $m$ and such that $gcd(m,n) = 1$, then such curves are "just" parametrised by the fibre product $X_1(n) \times_{X(1)} X_0(m)$ over the $j$-line. Computing these and doing something with them is a different story – Mummy the turkey Apr 09 '21 at 13:38

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This question inspired the work my advisor and I did together over the last couple of years

https://arxiv.org/abs/2001.05616

https://arxiv.org/abs/2104.01128

Garen Chiloyan
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