The Birch and Swinnerton-Dyer conjecture predicts that we have non-torsion rational points on an elliptic curve iff the L-function corresponding to the elliptic curve vanishes at 1. Thus BSD predicts that a positive integer N is the sum of two cubes if L(E_N, 1)=0, where L(E_N, s) is the L-function corresponding to the elliptic curve E_N: x^3+y^3=N. This offers a criterion for when the integer N is the sum of two cubes. Furthermore, when L(E_N, 1) is nonzero we get a formula for the number of elements in the Tate-Shafarevich group. Similarly, we consider certain families of sextic twists of the elliptic curve y^2=x^3+1 that are not defined over Q, but over Q[sqrt(-3)] and compute a formula that relates the value of the L-function L(E_D, 1) to the trace of a modular function at a CM point.

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