 These surfaces contain an elliptic curve (the image of the "x"-axis) and if formula_12 the image of the "y"-axis is a second elliptic curve.When formula_12, the Hopf surface is an elliptic fiber space over the projective line if 
formula_16 for some positive integers "m" and "n", with the map to the projective line given by formula_17, and otherwise the only curves are the two images of the axes.The Picard group of any primary Hopf surface is isomorphic to the non-zero complex numbers formula_18.