Let $(A,m)$ be a local commutative associative algebra over the field of complex numbers, $m^n\ne 0$, $m^{n+1}=0$ for some $n>0$, and

(1) $A$ is finite dimensional as vector space

(2) for any nonzero ideal $I$ of $A$, we have $m^n\subset I$

What can we say about such an $A$? For example, whether it is always a quotient algebra of some $\mathbb C[t^{m_1},\ldots,t^{m_k}]$?

I wonder whether there are some results charactered this kind of algebra more explicitly