KMO 2002 Problem 1

For a prime $p$ of the form $12k+1$ and $\mathbb{Z}_p=\{0,1,2,\cdots,p-1\}$ , let $$\mathbb{E}_p=\{(a,b) \mid a,b \in \mathbb{Z}_p, p\nmid 4a^3+27b^2\}$$ For $(a,b), (a',b') \in \mathbb{E}_p$ we say that $(a,b)$ and $(a',b')$ are equivalent if there is a non zero element $c\in \mathbb{Z}_p$ such that $p\mid (a' -ac^4)$ and $p\mid (b'-bc^6)$ . Find the maximal number of inequivalent elements in $\mathbb{E}_p$ .