Fields of definition of elliptic curves with prescribed torsion

Peter Bruin, Filip Najman
2017 Acta Arithmetica  
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