The general nonsingular projective cubic is of the form
$$F_\lambda = Y^2 Z - X (X - Z) (X - \lambda Z), \qquad \lambda \ne 0,1$$
where $F_\lambda$ and $F_\mu$ are isomorphic if they have the same modulus
$$J(\lambda) = \frac {27} 4 \frac {(1 - \lambda + \lambda^2)^2} {\lambda^2 (1 - \lambda)}$$
From this description, I expect the space of nonsingular cubics to be itself a curve (it has a single parameter!), but what kind of curve is it? So far, I can only tell that $\mathbb A^1 - \{ 0,1 \}$ is a 6-fold cover of it.