  This paper, motivated by problems in Diophantine analysis which can be
formulated as problems of finding rational points on the intersection of two
quadrics, presents an explicit construction of a rationally defined isomorphism
(biregular mapping) between a rationally defined smooth intersection of two
quadrics in projective three-space and an elliptic curve in Weierstrass form
which maps a distinguished rational point to the point at infinity. The usual
approach of transforming a smooth plane cubic to a curve in Weierstrass form by
mapping an inflection point to the point at infinity in a particular way is not
applicable in our setting, because there may be no inflection point defined
over the rationals. This difficulty is overcome by a construction dating back
to Nagell. The results are exemplified in two situations of number-theoretical
interest: Euler's problem of concordant forms and the occurrence of four
rational squares in arithmetic progressions.
