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We prove that all elliptic curves over quadratic fields with a subgroup isomorphic to C16, as well as all elliptic curves over cubic fields with a subgroup isomorphic to C2 × C14, are base changes of elliptic curves defined over Q. We obtain these results by studying geometric properties of modular curves and maps between modular curves, and then obtaining a modular description of these curves and maps.doi:10.4064/aa170323-20-9 fatcat:q7xxfnuz2ff6flrojppqtaozmq