On rational Frobenius Manifolds of rank three with symmetries

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.