In Classification of commutative Frobenius algebras , Jeremy Rickard showed that there are infinitely many commutative (local without loss of generality) Frobenius algebras of vector space dimension 14 over an infinite field.
Question 1: What is the smallest integer $l$ such that there are infinitely many commutative Frobenius algebras up to isomorphism of dimension $l$ (does this depend on the field?)?
In case this is field independent it might be interesting to find all such algebras of a given dimension $\leq l-1$ to do some tests (as there are some open problems on such algebras). This motivates the next question:
Question 2: Is there a quick way, using QPA, to obtain all finite dimensional local quiver algebras whose quiver has $r \geq 2$ loops of a given (small) vector space dimension over a finite field (with lets say 2 or 3 elements)?
This might be used to give a classification of Frobenius algebras over the field with 2 elements for small dimensions.