Computing Canonical Heights With Little (or No) Factorization

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