Existence of a family of elliptic curves with large torsion subgroup 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? 
 A: One reason might be that these families are the least dense.
See Table 1 of http://www-personal.umich.edu/~asnowden/papers/torsion-112013.pdf
Or maybe I will just copy it here. $T$ is the torsion group, $d$ is the their $d$ (reciprocal of power density), and $r$ is the rank record from Dujella. I don't know if one could argue that these 4 remaining families are so sparse that one should not expect an infinite family.

A: It might be worth noting that in the following paper, which is also in Dujella's list of references,
A. O. L. Atkin and F. Morain, Finding suitable curves for the elliptic curve method of factorization, Math. Comp. 60 (1993) (available here: http://www.ams.org/journals/mcom/1993-60-201/S0025-5718-1993-1140645-1/S0025-5718-1993-1140645-1.pdf)
A. O. L. Atkin and F. Morain exhibits families of rank $ \geq 1$ and torsion group
$\mathbb Z/8\mathbb Z \times \mathbb Z/2\mathbb Z$, $\mathbb Z/5\mathbb Z$, $\mathbb Z/7\mathbb Z$, $\mathbb Z/9\mathbb Z$ and $\mathbb Z/10\mathbb Z$.
However, their families are not $1$-parameter families, i.e. curves over $\mathbb Q(T)$, but rather parametrized by points on elliptic curves. Whether or not that adds anything to the likelihood of finding $1$-parameter families of rank $\geq 1$ for those less frequent torsion groups, I do not know.
