  We study Frobenius manifolds of rank three and dimension one that are related
to submanifolds of certain Frobenius manifolds arising in mirror symmetry of
elliptic orbifolds. We classify such Frobenius manifolds that are defined over
an arbitrary field $\mathbb{K} \subset \mathbb{C}$ via the theory of modular
forms. By an arithmetic property of an elliptic curve $\mathbb{E}_\tau$ defined
over $\mathbb K$ associated to such a Frobenius manifold, it is proved that
there are only two such Frobenius manifolds defined over $\mathbb C$ satisfying
a certain symmetry assumption and thirteen Frobenius manifolds defined over
$\mathbb Q$ satisfying a weak symmetry assumption on the potential.
