  We give a new method of producing relations in the tautological ring R(M_{g,
1}), using the sl_2-action on the Chow ring of the universal Jacobian. With
these relations, we prove that R(M_{g, 1}) is generated by \kappa_i for i no
greater than g/3, together with \psi. Our computation shows that Faber's
conjectures for M_{g, 1} are true for g up to 19. Further, by pushing relations
forward to M_g, we obtain a new proof of Faber's conjectures (for M_g) for g up
to 23. For g = 24, our method recovers all the Faber-Zagier relations. We also
give an algebraic proof of an identity of Morita.
