How can you show that if the characteristic of an elliptic curve $y^2 = x^3 + ax + b$ is 2 or 3 the equation fails? For characteristic 2 I know the equation must be written as $y^2 + ay = x^3 + bx^2 + cxy + dx + f$ but I'm not sure how to show it fails in the first equation.
I am led to believe it has something to do with modulus 2 and the determinant $\Delta=-16(4a^3 + 27b^2)$.