Let $S$ be an elliptic surface over $\mathbb{C}$, i.e. a smooth, projective algebraic surface equipped with a morphism $f: S \to C$ to a curve $C$ such that the generic fibre is an elliptic curve over $\mathbb{C}(C)$. The Kodaira dimension of an elliptic surface is at most $1$ (but can be $0$ or $-\infty$). If one asks that the genus of $C$ satisfies $g(C) \geq 2$, does this force the Kodaira dimension to be equal to $1$, or can it still be $0$ or $-\infty$?