Addendum to a paper of Harada and Lang.

*(English)*Zbl 0759.11021[Cf. K. Harada and M. L. Lang, J. Algebra 125, 298-310 (1989; Zbl 0715.11029).]

The usual proof, by Riemann-Roch, that an elliptic curve can be put into Weierstraß form does not provide a practical algorithm, but appropriate transformations are described, e.g., by J. W. S. Cassels [Lectures on Elliptic Curves, Cambridge Univ. Press (1991; Zbl 0752.14033)] for general cubics, quartics, and intersections of two quadric surfaces (with a rational point in each case). The present paper discusses such algorithms for the first two cases. [Author’s note: Both these algorithms are included as procedures in “apecs” (arithmetic of plane elliptic curves) which is a comprehensive program written in MAPLE intended for free distribution].

The usual proof, by Riemann-Roch, that an elliptic curve can be put into Weierstraß form does not provide a practical algorithm, but appropriate transformations are described, e.g., by J. W. S. Cassels [Lectures on Elliptic Curves, Cambridge Univ. Press (1991; Zbl 0752.14033)] for general cubics, quartics, and intersections of two quadric surfaces (with a rational point in each case). The present paper discusses such algorithms for the first two cases. [Author’s note: Both these algorithms are included as procedures in “apecs” (arithmetic of plane elliptic curves) which is a comprehensive program written in MAPLE intended for free distribution].

Reviewer: H.G.Quebbemann (Oldenburg)

##### MSC:

11G05 | Elliptic curves over global fields |

11Y16 | Number-theoretic algorithms; complexity |

14H52 | Elliptic curves |

14Q05 | Computational aspects of algebraic curves |

##### Software:

Maple
Full Text:
DOI

##### References:

[1] | Hancock, H, Lectures on the theory of elliptic functions, (1958), Dover New York |

[2] | Harada, K; Lang, M.L, Some elliptic curves arising from the Leech lattice, J. algebra, 125, 298-310, (1989) · Zbl 0715.11029 |

[3] | Mordell, L.J, Diophantine equations, (1969), Academic Press London · Zbl 0188.34503 |

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