Explicit Result on Equivalence of Rational Quadratic Forms Avoiding Primes

Given a pair of regular quadratic forms over \(\mathbb{Q}\) which are in the same genus and a finite set of primes \(P\), we show that there is an effective way to determine a rational equivalence between these two quadratic forms which are integral over every prime in \(P\). This answers one of the principal questions posed by Conway and Sloane in their book {\em Sphere packings, lattices and groups}, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol 290, Springer-Verlag, New York, 1999; page 402.