On standardized models of isogenous elliptic curves
HTML articles powered by AMS MathViewer
- by Samir Siksek PDF
- Math. Comp. 74 (2005), 949-951 Request permission
Abstract:
Let $E,~E^\prime$ be isogenous elliptic curves over ${\mathbb Q}$ given by standardized Weierstrass models. We show that (in the obvious notation) \[ a_1’ =a_1, \qquad a_2’ = a_2, \qquad a_3’ = a_3 \] and, moreover, that there are integers $t,~w$ such that \[ a_4’ = a_4 - 5t~\text {and}~ a_6’ = a_6 - b_2 t - 7w, \] where $b_2=a_1^2+4 a_2$.References
- B. J. Birch and W. Kuyk (eds.), Modular functions of one variable. IV, Lecture Notes in Mathematics, Vol. 476, Springer-Verlag, Berlin-New York, 1975. MR 0376533
- John Cannon and Derek Holt (eds.), Computational algebra and number theory, Elsevier Ltd, Oxford, 1997. J. Symbolic Comput. 24 (1997), no. 3-4. MR 1484477
- J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 1628193
- Michiel Hazewinkel, Formal groups and applications, Pure and Applied Mathematics, vol. 78, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506881
- Taira Honda, Formal groups and zeta-functions, Osaka Math. J. 5 (1968), 199–213. MR 249438
- Taira Honda, On the theory of commutative formal groups, J. Math. Soc. Japan 22 (1970), 213–246. MR 255551, DOI 10.2969/jmsj/02220213
- Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
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
- 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).
- © Copyright 2004 American Mathematical Society
- Journal: Math. Comp. 74 (2005), 949-951
- MSC (2000): Primary 11G05
- DOI: https://doi.org/10.1090/S0025-5718-04-01690-4
- MathSciNet review: 2114657