I know what is the $j$-invariant but I am asking about a general definition in sence of classical invariant theory. The following possible definition seems to be wrong: Consider an elliptic curve $C: y^2=x^3+ax+b.$ Acting by the transformation $$ \begin{array}{c} x=a_{1,1}x'+a_{1,2}y'+c_1,\\ y=a_{2,1}x'+a_{2,2}y'+c_2, \end{array} $$ we get new curve $C'.$ The invariant is a function of coefficients of the curves which is stable under above transformation. This definition is wrong becouse $C'$ now consists $y'^3$ and it isn't elliptic curve.

So, what is correct definition of an invariant of elliptic curve?