| We investigate computational problems related to modular parametrizations of elliptic curves defined over Q. We develop algorithms to compute the Mazur Swinnerton-Dyer critical subgroup of elliptic curves, and verify that for all elliptic curves of rank two and conductor less than a thousand, the critical subgroup is torsion. We also develop algorithms to compute Fourier expansions of Gamma0(N )-newforms at cusps other than the cusp at infinity. In addition, we study properties of Chow-Heegner points associated to a pair of elliptic curves. We proved that the index of Chow-Heegner points are always divisible by two when the conductor N has many prime divisors, .We also develop an algebraic algorithms to compute the Chow-Heegner points. |
