Let $X$ be a threefold with Kodaira dimension 2 such that the Iitaka map $\Phi :X \to Y$ is not isotrivial. The generic fiber of $\Phi$ is an elliptic curve. 

__Q1.__ How many such threefolds exist, and how many explicit examples can be given? 

__Q2.__ Let $\rho : T_Y \to H^1(X_y, T_{X_y})$ be the Kodaira-Spencer map (the differential of the moduli map $\mu : Y \to \mathcal{M}$, where $\mathcal{M}$ is the moduli space of elliptic curves). Is $\rho$ generically injective?