Fields of definition of elliptic curves with prescribed torsion
Peter Bruin, Filip Najman

TL;DR
This paper proves that certain elliptic curves over quadratic and cubic fields with specific torsion subgroups are actually base changes of elliptic curves defined over the rationals, using modular curve analysis.
Contribution
It provides a modular curve-based proof that elliptic curves with specific torsion over quadratic and cubic fields are base changes from .
Findings
Elliptic curves over quadratic fields with C_{16} torsion are base changes from .
Elliptic curves over cubic fields with C_2 times C_{14} torsion are base changes from .
The study of modular curves and maps yields these classification results.
Abstract
We prove that all elliptic curves over quadratic fields with a subgroup isomorphic to , as well as all elliptic curves over cubic fields with a subgroup isomorphic to , are base changes of elliptic curves defined over . 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.
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