Computing canonical heights with little (or no) factorization

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