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This paper, motivated by problems in Diophantine analysis which can beformulated as problems of finding rational points on the intersection of twoquadrics, presents an explicit construction of a rationally defined isomorphism(biregular mapping) between a rationally defined smooth intersection of twoquadrics in projective three-space and an elliptic curve in Weierstrass formwhich maps a distinguished rational point to the point at infinity. The usualapproach of transforming a smooth plane cubic to a curve in Weierstrass form bymapping an inflection point to the point at infinity in a particular way is notapplicable in our setting, because there may be no inflection point definedover the rationals. This difficulty is overcome by a construction dating backto Nagell. The results are exemplified in two situations of number-theoreticalinterest: Euler's problem of concordant forms and the occurrence of fourrational squares in arithmetic progressions.

Explicit transformation of an intersection of two quadrics to an elliptic curve in Weierstrass form