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On standardized models of isogenous elliptic curves

Author: Samir Siksek
Journal: Math. Comp. 74 (2005), 949-951
MSC (2000): Primary 11G05
Published electronically: July 7, 2004
MathSciNet review: 2114657
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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 $t,~w$ such that

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

where $b_2=a_1^2+4 a_2$.

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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

Keywords: Elliptic curves, formal groups, isogenies, standardized models
Received by editor(s): November 8, 2003
Published electronically: July 7, 2004
Additional Notes: The author’s work is funded by a grant from Sultan Qaboos University (IG/SCI/DOMS/02/06).
Article copyright: © Copyright 2004 American Mathematical Society