  Let C be a complex curve of genus g, let J(C) be its Jacobian and let R(C) be
its tautological ring, that is, the group of algebraic cycles modulo algebraic
equivalence. We study the algebraic structure of R(C). In particular, we give a
detailed description of all the possibilities that may occur for g<9: we
construct convenient basis and we determine the matrices representing the
Fourier transform and both intersection and Pontryagin products explicitly. In
particular, we estimate the dimension of R(C).
