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On standardized models of isogenous elliptic curves

Author(s): Samir Siksek.
Journal: Math. Comp. 74 (2005), 949-951.
MSC (2000): Primary 11G05
Posted: July 7, 2004
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Abstract: Let $E,~E^\prime$ be isogenous elliptic curves over ${\mathbb Q}$ given by standardized Weierstrass models. We show that (in the obvious notation)

$\begin{displaymath}a_1^{\prime}=a_1, \qquad a_2^{\prime} = a_2, \qquad a_3^{\prime} = a_3 \end{displaymath}$

and, moreover, that there are integers

such that

$\begin{displaymath}a_4^{\prime} = a_4 - 5t~\text{and}~ a_6^{\prime} = a_6 - b_2 t - 7w, \end{displaymath}$

where

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3.: J.E. Cremona, Algorithms for Modular Elliptic Curves (second edition), Cambridge University Press, 1996. MR 99e:11068
4.: M. Hazewinkel, Formal Groups and Applications, Academic Press, 1978. MR 82a:14020
5.: T. Honda, Formal Groups and Zeta-Functions, Osaka J. Math. 5 (1968), 199-213.MR 40:2683
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7.: J.H. Silverman, The Arithmetic of Elliptic Curves, GTM 106, Springer-Verlag, 1986. MR 87g:11070

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Additional Information:

Samir Siksek
Affiliation: Department of Mathematics and Statistics, College of Science, P.O. Box 36, Sultan Qaboos University, Al-Khod 123, Oman
Email: siksek@squ.edu.om

DOI: 10.1090/S0025-5718-04-01690-4
PII: S 0025-5718(04)01690-4
Keywords: Elliptic curves, formal groups, isogenies, standardized models
Received by editor(s): November 8, 2003
Posted: July 7, 2004
Additional Notes: The author's work is funded by a grant from Sultan Qaboos University (IG/SCI/DOMS/02/06).
Copyright of article: Copyright 2004, American Mathematical Society


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