In "On elliptic modular surfaces", Shioda proves some interesting theorems on smooth elliptic surfaces (admitting a section); he then focuses on "modular elliptic surfaces" and proves some more theorems, e.g. he shows that there is a one to one correspondence between modular forms vanishing at cusps of a modular curve, and holomorphic two forms on the corresponding modular surfaces.

After this short introduction, my question is: Can you give me an explicit example of a smooth elliptic surface (with section) with at least 3 singular fibers, which is not modular? What if the base curve has genus zero?

More explanation: Let $S\to B$ be an elliptic surface with section and at least 3 singular fibers. Let $\Sigma\subset B$ be the complement of singular values. Then $\Sigma=\mathbb{H}/G$, is a quotient of upper half plane. The $j$ function of this elliptic fibration $j:\Sigma \to \mathbb{H}/PSL(2,\mathbb{Z})$ (is it always holomorphic?) gives a map $$ \iota: G \to SL(2,\mathbb{Z}).$$ I then feel that $S$ should be the modular surface corresponding to $\iota(G)\subset SL(2,\mathbb{Z})$ and I can't see what might go wrong?!

Regarding the answer of Remke below: Assuming that the map $\iota$ above has finite dimensional kernel, it seems that $S$ should be the pull-back of the modular surface corresponding to $\iota(G)$ via the ramified covering map $$ \mathbb{H}/G \to \mathbb{H}/\iota(G);$$ is this true?