The Langlands Programme predicts that a weight 2 newform f over a number eld K with integer Hecke eigenvalues generally should have an associated elliptic curve Ef over K. In [GMS14], we associated, building on works of Darmon [Dar01] and Greenberg [Gre09], a p-adic lattice to f, under certain hypothesis, and implicitly conjectured that is commensurable with the p-adic Tate lattice of Ef . In this paper, we present this conjecture in detail and discuss how it can be used to compute, directly from f, a Weierstrass equation for the conjectural Ef . We develop algorithms to this end and implement them in order to carry out extensive systematic computations in which we computeWeierstrass equations of hundreds of elliptic curves, some with huge heights, over dozens of number elds. The data we obtain give extensive support for the conjecture and furthermore demonstrate that the conjecture provides an ecient tool to building databases of elliptic curves over number elds.