3and henceabcd=αγβδabcdαβγδ.This can be also expressed by saying that the formfis obtained from the formfby using the change of variablesx→αx+βy,y→γx+δy.We write this in the formf=Mf.According to Lagrange two binary quadratic formsfandgare calledequivalentif one transforms to another under the change of variables as above defined byan integral matrix with determinant±1. An equivalence class is called theclassof forms. Obviously, for anyn∈Z, the set of integral solutions of the equationsf(x, y) =ndepends only on the class of forms to whichfbelongs. Also it isclear that two equivalent forms have the same discriminant.1.2As we saw before any lattice Λ determines a class of forms expressing thedistance from a point in Λ to the origin. Conversely, given a positive definitebinary formf=ax2+ 2bxy+cy2we can find a lattice Λ corresponding tothis form. To do this we choose any vectorvof length√aand letwbe the
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