A Weierstrass equation of the form $y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$ over some field is called elliptic curve if the discriminant $\triangle \neq 0$. If the characteristic of the field is not equal to 2 and 3, then the above elliptic curve is transformed to the normal form $y^2=x^3+ax+b$ and the discriminant $\triangle =4a^3+27b^2 \neq 0$.
My question is that, can we write an elliptic curve in normal form over some finite field with characteristic 2 or 3 and having nonzero discriminant.
For instance, $E:y^2=x^3+x+1$ over $\mathbb{F}_3$. Here the discriminant $\triangle =4a^3+27b^2 = 1 (\text{mod } 3) \neq 0$.
Is it correct or wrong? Am I making any mistake somewhere?