Section Three-Descent, which concerns three descent on elliptic curves, is relevant for studying rational points on plane cubics (genus one models of degree 3) over the rationals.
A very efficient point search for schemes is available; for details see Section Searching for Points. This will find all rational points up to absolute height H on the curve associated to a given genus one model in time O(H2/d), where d is the dimension of the ambient projective space.
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