For the Dedekind eta function, defined as usual by $\eta(q) = q^{\frac1{24}} \prod\limits_{n=1}^{\infty} (1-q^{n})$, let for brevity $e_k:=\eta(q^k)$.

An **eta product identity** (or **eta identity** for short) is then defined as a homogenous polynomial in the $e_k$ with integer coefficients such that its Taylor series vanishes identically. There are already more than 6300 of them in Michael Somos' collection.

Asking bluntly like that how many eta identities exist, the answer is of course infinitely many, as they form a (graded) vector space.So we may try, as does Somos, to restrict to "irreducible" ones in an appropriate sense. Now it turns out that it is a bit tricky to come up with a good definition of irreducibility for an eta identity. Some requirements are obvious:

- it should be irreducible as a polynomial, including that
- not only the $\gcd$ of the coefficients (up to sign) should be $1$, but also the $\gcd$ of the eta exponents (i.e. the indices of the $e_k$'s) involved
- it should not be expressible as the sum of two shorter eta identities
- if there are several linear independant eta identities with given level, degree and rank, we should only count $k$ basis elements where $k$ is the dimension of the vector space generated by them, all others would be considered "reducible in terms of this basis".

A (minor) problem can occur if we add two identities whose monomials are not all disjoint in a way that some monomials cancel out. In particular, we can take any pair of 3-term identities, say $a+b+c$ and $d+e+f$, then $d(a+b+c)-a(d+e+f)=bd+cd-ae-af$ is a 4-term identity. Should we call it reducible? Some 4-term identities in Somos' collection can be obtained like that, but many can't (a priori). There are also pairs of 4-term identities of the form $a+b+c+d$ and $a+b+kc+e$ with $k\in\mathbb Q$, so their difference yields a 3-term identity, but as it is made up of two longer identities, there would be no reason anyway to call that one reducible.

So it seems best to stick to the four above criteria for an irreducible eta identity, and not to worry about some 4-term identities that would fall through.

There are still infinitely many irreducible eta identities, because e.g. all Schläfli type modular equations can be written in terms of eta products, so (irreducible) eta identities exist at least for all levels divisible by 4.

Now, many other types of modular equations, like e.g. theta functions that can be expressed as eta quotients, tend to come in *finite* quantities, and I think that we can impose feasible constraints on eta identities such that there are only finitely many fulfilling those constraints. So which constraints might that be? Some questions below are stronger than others.

For a given level $N$, are there only finitely many irreducible eta identities?(Note that if length and degree are limited, that follows from the existence of the Sturm bound)

- (more or less equivalent to 1.)
For a given level $N$, is there a maximal length and/or a maximal degreeof an irreducible eta identity?

It seems like for a given level, the identities of maximal degree tend to have more symmetries, more precisely, they are most often self-dual in the sense defined here, or sometimes, even stronger, each term is self-dual. (I am aware that the collection is not necessarily exhaustive even for rather small levels, so this may be a wrong impression).

Are there only finitely many 3-term identities(all levels combined)?

- If so, then it makes also sense to ask:
for a given length $k>3$, are there only finitely many irreducible $k$-term identities?

Are there irreducible eta identities for each level $N$that is not prime?

Concerning the last question above: it seems like for odd $N$'s, there are relatively few identities, and the bigger the minimal prime factor, the more intricate they are. E.g. for level $13^2$, according to Somos the "smallest example has 50 terms of degree 14 and rank 182", and for level $11^2$, the smallest example has 156 terms of degree 60 and rank 660. But it does exist!

On the other hand, if $N$ has only small prime factors, there are lots of identities. For $N=12$ alone, the collection lists over 1000, the vast majority of them with only four terms.

* EDIT:* Subsequently to Jeremy's answer, Michael Somos has added a section with the (unique) identities of level $p^2$ for primes $p\le 19$. For the record: some of their parameters (summarized from the comments) are $$\matrix{p&|°ree&rank& \#\ terms\cr\hline 2&|&24&48&3 \cr 3&|&12&36&4\cr5&|&6&30&6\cr 7&|&8&56&11\cr 11&|&60&660&156\cr 13&|&14&182&50 \cr 17&|&72&1224&421 \cr 19&|&60&1140&436 }$$