On the computation of Mordell-Weil and 2-Selmer groups of elliptic curves. (Q1414957)

Let \(E\) be an elliptic curve defined over \(\mathbb Q\) with discriminant \(\Delta\); if \(E\) has three rational points \((e_j,0)\) (\(j = 1, 2, 3\)) of order \(2\), then the map sending a point \(P = (x,y)\) of order \(> 2\) to the triple of elements \((x-e_j)\mathbb Q^{\times\,2}\) induces an injective homomorphism \(\alpha: E(\mathbb Q)/2E(\mathbb Q) \longrightarrow (\mathbb Q^\times/\mathbb Q^{\times\,2})^3\). This monomorphism can be used to verify the independence of rational points in \(E(\mathbb Q)\). For general elliptic curves, Brumer suggested to compose \(\alpha\) with reduction modulo (prime ideals above) \(p\) for primes not dividing \(6 \Delta\). Using sufficiently many such primes \(p\), one gets an injective homomorphism \(\varepsilon: E(\mathbb Q)/2E(\mathbb Q) \longrightarrow (\mathbb Z/2\mathbb Z)^M\) for some \(M\) which can be used to check whether a given list of rational points is independent.