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: <math> \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} </math>
has integer entries and determinant 1, the map <math> f(x,y) \mapsto f(\alpha x + \beta y, \gamma x + \delta y)</math> is a (right) [[group action]] of <math>\mathrm{SL}_2(\mathbb{Z})</math> on the set of binary quadratic forms. The equivalence relation above then arises from the general theory of group actions.
If <math>f=ax^2+bxy+cy^2</math>, then important invariants include
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