Is there a computer programm that can list all finite dimensional commutative local (nonsemisimple) k-algebras $K[x_1,x_2,...,x_n]/I$ over a finite field k (of order $q$) with $J^l=0$ for a given $q, n$ and $l$ , when $J=\langle x_1,...,x_n\rangle$ denotes the Jacobson radical and $I \subseteq J^2$?

One way might be as follows: Let $A$ be the algebra $K[x_1,x_2,...,x_n]/J^l$, then one just has to find all ideals $I$ (submodules) of $A$ and look at the quotients.

Is there an estimate how many there are (up to isomorphism or not)?

How and in what time can a computer check if two such algebras are isomorphic?

embedding dimensionof the algebra. You're asking for the algebras of given embedding dimension and socle degree. $\endgroup$ – Zach Teitler Jul 2 '18 at 8:34