Andrej Dujella's excellent web-site "High rank elliptic curves with prescribed torsion", at http://web.math.pmf.unizg.hr/~duje/tors/tors.html gives example of the (current) largest known rank of an elliptic curve over $\mathbb{Q}$ having each of the fifteen possible torsion subgroup structures.

The related web-site "Infinite families of elliptic curves with high rank and prescribed torsion", at http://web.math.pmf.unizg.hr/~duje/tors/generic.html, gives information on the existence of infinite families.

There are no known families with torsion

$\mathbb{Z} / 9\mathbb{Z}$, $\mathbb{Z} / 10\mathbb{Z}$, $\mathbb{Z} / 12\mathbb{Z}$, and $\mathbb{Z} / 2\mathbb{Z} \times \mathbb{Z} / 8\mathbb{Z}$.

I have already spent a reasonable amount of time hunting for such a beast. To save me wasting my time, I would like to ask experts whether they would expect such a family to exist, and if so, why? Alternatively, if they do not expect such a family to exist, could they also give a plausible reason?

andwith rank as high as the largest known rank of an elliptic curve over $\mathbb{Q}$ having the same torsion group $T$. $\endgroup$3more comments