Abstract. Let E/Q be an elliptic curve with discriminant ∆, and let P ∈ E(Q). The standard method for computing the canonical heightĥ(P ) is as a sum of local heightsĥ(P ) =λ∞(P) + pλ p(P ). There are well-known series for computing the archimedean heightλ∞(P ), and the non-archimedean heightsλp(P ) 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 nonarchimedean contribution toĥ(P ) which is quite practical and requires little or no factorization. We also give some numerical examples illustrating the algorithm.Let E be an elliptic curve defined over a number field K, say given by a Weierstrass equationThe canonical height on E is a quadratic formThe canonical height is an extremely important theoretical and computational tool in the arithmetic study of elliptic curves. See [18, Chapter VIII, Section 9] for the definition and basic properties ofĥ, and [20], [21], and [23] for some discussion of how to computeĥ in practice. In this paper, which may be considered as a continuation of our earlier note [20], we will discuss the computation of the canonical height for curves E whose coefficients a 1 , . . . , a 6 are large. We note that this is not a mere intellectual exercise, since curves with huge integer coefficients have already made their appearance in the search for curves whose Mordell-Weil group E(Q) has large rank [5], [11], [12], [13], [14], and the standard tool for proving that a set of points P 1 , . . . , P r ∈ E(Q) is linearly independent is to check the non-vanishing of the height regulator matrix det P i , P j . Here the height pairing · , · is defined (up to a normalizing factor) by the formulaTate's definitionĥ(P ) = lim n→∞ 4 −n h x(2 n P ) of the canonical height is not practical for numerical computations. Instead, one uses the Néron-Tate decomposition of the canonical height into a sum of local heights, one for each distinct