Abstract: The Birch and Swinnerton-Dyer (BSD) conjecture for elliptic curves defined over the rational numbers is a famous problem that has been open for over forty years and is one of the seven Millennium Prize problems. The BSD conjecture is considered to be the first nontrivial number theoretic problem put forth as a result of explicit machine computation --- in the late '50s at Cambridge University. This conjecture relates the rank of the Mordell-Weil group, the group of rational points of an elliptic curve, a quantity which seems to be difficult to pin down, to the order of vanishing of the L-function of the elliptic curve at its central point.

We present algorithmic and theoretical advances with regards to some of the terms appearing in the BSD formula, namely the sizes of the torsion subgroup and the Shafarevich-Tate group.

Firstly, we introduce an algorithm to compute elliptic curve torsion subgroup. The randomized version of this procedure runs in expected time which is essentially linear in the number of bits required to write down the equation of the elliptic curve.

Next, we discuss a conjecture of Hindry, who proposed a Brauer-Siegel type formula for elliptic curves. Driven by a suggestion of Hindry, we prove assuming standard conjectures that there are infinitely many elliptic curves with Shafarevich-Tate group of size about as large as the square root of the minimal discriminant of the curve. This improves on a result of de Weger.

Finally, we consider certain quartic twists of an elliptic curve. We establish a reduction between the problem of factoring integers of a certain form and the problem of computing rational points on these twists. We illustrate that the size of Shafarevich-Tate group of these curves will make it computationally expensive to factor integers by computing rational points via the Heegner point method.