Can the coefficients of an elliptic curve be recovered from the curve itself?

Question

Let $K$ denote an algebraically closed field. Suppose $E$ and $E'$ are elliptic curves given by Weierstrass equations with parameters $(a_1,a_3,a_2,a_4,a_6)$ and $(a_1',a_3',a_2',a_4',a_6').$ From $E = E'$ as subsets of projective space, can we deduce that $a_i = a'_i$?