 Elliptic surface of Kodaira dimension minus infinity

For an elliptic curve $E$, projection ${\mathbb{P}}^1\times E\to {\mathbb{P}}^1$ is an elliptic fibration. Its canonical bundle is the pullback of $K_{{\mathbb{P}}^1}$, so it has no nonzero pluricanonical sections and its Kodaira dimension is $-\infty$.