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Some families of Mordell curves associated to cubic fields. (English) Zbl 1080.14520

Summary: We introduce some Mordell curves of two different natures both of which are associated to cubic fields. One set of them consists of those elliptic curves whose rational points over the rational number field are described by or closely related to cubic fields. The other is a one-parameter family of Mordell curves which gives all (cyclic) cubic twists and all quadratic twists of the Fermat curve \(X^3+Y^3+Z^3=0\).

MSC:

11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11G05 Elliptic curves over global fields
14H25 Arithmetic ground fields for curves
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