Generally speaking, I am interested in counting the number of $\mathbb{F}_p$- isomorphism classes of elliptic curves containing a specific torsion subgroup, and I was wondering if there were any simple formulas for particular torsion subgroups. Any information on this topic is appreciated.

More specifically, I would like to count the number of $\mathbb{F_p}$-isomorphism classes of elliptic curves whose torsion subgroup contains $\mathbb{Z}/N \times \mathbb{Z}/N$, where $N$ is a small fixed integer, for instance one for which the modular curve $X(N)$ has genus 0.

I know that if you fix a prime $p$, and fix an isogeny class of such curves over $\mathbb{F}_p$, then Schoof (in "Nonsingular plane cubic curves over finite fields") has formulas which express the number of isomorphism classes of these curves in the given isogeny class in terms of class numbers of certain quadratic extensions of the rationals. So, for example, summing these expressions over all possible isogeny classes is an answer to the question. However, I don't know of any easy way to compute this sum of class numbers, and it seems that such a sum could potentially be expressed more simply.

Also, Schoof's result holds for all $N$, and I thought it might be possible for smaller $N>1$'s that things simplified a bit.

Thanks for the help!