Computing Canonical Heights With Little (or No) Factorization

Let be an elliptic curve with discriminant , and let . The standard method for computing the canonical height is as a sum of local heights . There are well-known series for computing the archimedean height , and the non-archimedean heights are easily computed as soon as all prime factors of have been determined. However, for curves with large coefficients it may be difficult or impossible to factor . In this note we give a method for computing the non-archimedean contribution to which is quite practical and requires little or no factorization. We also give some numerical examples illustrating the algorithm.