Determine if a curve is an elliptic curve given characteristic 2 and 3?

Question

Given $E:$ $y^2 = x^3 + a$, $a$ being an element of the field $K$, determine if $E$ if an elliptic curve if:

1. The characteristic of $K$ is not 2 or 3
2. The characteristic of $K$ is 2
3. The characteristic of $K$ is 3

I know definition of a characteristic but as a beginner in the introduction to elliptic curves I do not understand how characteristic apply to the algebraic equation of the type $E$ above.

I know if $char(K) = 2$, then $2=0$ but I don't know where to go from here. I also know that a curve is elliptic if the partial derivatives are not simultaneously null but Im failing to understand where the characteristic plays part.