  The Langlands Programme predicts that a weight 2 newform f over a number
field K with integer Hecke eigenvalues generally should have an associated
elliptic curve E_f over K. In our previous paper, we associated, building on
works of Darmon and Greenberg, a p-adic lattice to f, under certain hypothesis,
and implicitly conjectured that this lattice is commensurable with the p-adic
Tate lattice of E_f . In this paper, we present this conjecture in detail and
discuss how it can be used to compute, directly from f, an explicit Weierstrass
equation for the conjectural E_f . We develop algorithms to this end and
implement them in order to carry out extensive systematic computations in which
we compute Weierstrass equations of hundreds of elliptic curves, some with huge
heights, over dozens of number fields. The data we obtain provide overwhelming
amount of support for the conjecture and furthermore demonstrate that the
conjecture provides an efficient tool to building databases of elliptic curves
over number fields.
