How does one compute the modular group of the torus? I see how Dehn twists generate the modular group, and I see how Dehn twists are really automorphisms of isotopy classes. Based on this, it seems that the modular group should be Aut(pi1(T^2))=Aut(Z^2)=GL(2,Z). But I've read that the modular group is in fact SL(2,Z). How does this work? I may have something to do with orientation-preservation, but I haven't been able to flesh this out. Thanks in advance.