The first provides an alternative to computing the height pairing matrix of the given set of points and showing that its determinant is non-zero. While that is easily done, for curves of large rank it requires some delicate consideration of precision in order to be sure of the result. The method here, by contrast, involves only “discrete” computations: finding roots of cubics and evaluating quadratic characters modulo primes. It was also described by Silverman in [6], attributed there to Brumer and myself. In fact, Brumer described the method to me in 1996; it was apparently used by him and Kramer in verifying the examples in [2], though the method is not explicitly mentioned there; so the method goes back to 1975 at least. We give it here as it is closely related to, and leads to, our second section where we apply similar ideas to 2-Selmer groups. We illustrate the method with the Martin-McMillen curve which has 23 independent points. The second problem arises when doing explicit 2-descents on elliptic curves with no 2-torsion, as implemented in our program mwrank. Following the method set out in [1], we represent elements of the Selmer group S(E/Q) by quartics g(X) ∈ Z[X] such that the genus 1 curve Y 2 = g(X) is a 2-covering of E. These quartics are found by a finite search procedure. In [1], the resulting set of (equivalence classes of) quartics is treated as a set, without making explicit its structure as an elementary abelian 2-group. Indeed, one check on the calculations is to make sure that the size of the set obtained is a power of 2. We will show how to make explicit use of the group structure on S(E/Q), via a homomorphism to (Z/2Z) for some M > 0. This has a number of practical advantages in terms of the running time of the resulting algorithm: we do not need to check equivalences between the quartics found; for a curve of rank r, we only find and consider r quartics instead of 2, saving much time in the search for rational points on the associated 2-coverings; and the search itself is made faster, since for every quartic we find, the remaining part of the search region is reduced by a factor of 2.