If I want to know properties of $E_{p+1}$ modulo $p$, do you know a name for this modular form, so that it is easier to search via the internet?
So far, what I know is that $E_{p-1}$ is the Hasse invariant, which is connected to supersingular elliptic curves, and that $E_{p+1}$ is connected to $E_{p-1}$ via the Serre-Ramanujan differential operator $\partial$, where we have the identities \begin{align*} \partial E_{p-1} &= E_{p+1}\\ \partial E_{p+1} &= -E_4 E_{p-1} \end{align*} when working modulo $p$ (and these identities are used to show things like, if $A\in\mathbb{F}_{p}[X,Y]$ is the polynomial such that $A(E_4,E_6)=E_{p-1}$ and $B$ is likewise for $\mathbb{F}_{p}[X,Y]$, then $A$ has no repeated factors and is relatively prime to $B$). However, for a nonexpert like me, it took some digging around until I finally realized $E_{p-1}$ is also the Hasse invariant (and so you could use tools from elliptic curves as well). Is there also a different name for $E_{p+1}$ $\pmod{p}$?
Unfortunately, the only name I know for this besides $E_{p+1}$ is "$B$", which is not so useful for googling. Serre's article on modular forms mod $p$ is a standard reference for this sort of thing, by the way.