= Unit square-class bound for two-isogeny descent
Suppose the <ring of integers of a number field> is a <principal ideal domain>, $a,b$ are integral, and $b$ is a <unit>. On $y^2=x(x^2+ax+b)$ every nonzero $x$-coordinate has even <valuation> at every finite <prime>: positive valuation gives $2v(y)=v(x)$, and negative valuation gives $2v(y)=3v(x)$. Thus the <two-torsion square-class homomorphism> takes values among unit <square classes>. For the <Gaussian integers>, their <unit group> modulo squares has two elements, represented by $1,i$. This can reduce a <two-isogeny descent> to a very small calculation even though the field contains no ordering.
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