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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

High rank elliptic curves with torsion group $\mathbb{Z} /(2\mathbb{Z} )$


Authors: Julián Aguirre, Fernando Castañeda and Juan Carlos Peral
Journal: Math. Comp. 73 (2004), 323-331
MSC (2000): Primary 11Y50
Published electronically: May 30, 2003
MathSciNet review: 2034125
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Abstract: We develop an algorithm for bounding the rank of elliptic curves in the family $y^2=x^3-B\,x$, all of them with torsion group $\mathbb{Z} /(2\,\mathbb{Z} )$and modular invariant $j=1728$. We use it to look for curves of high rank in this family and present four such curves of rank $13$ and $22$ of rank $12$.


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Additional Information

Julián Aguirre
Affiliation: Departamento de Matemáticas, Universidad del País Vasco, Aptdo. 644, 48080 Bilbao, Spain
Email: mtpagesj@lg.ehu.es

Fernando Castañeda
Affiliation: Departamento de Matemáticas, Universidad del País Vasco, Aptdo. 644, 48080 Bilbao, Spain
Email: mtpcabrf@lg.ehu.es

Juan Carlos Peral
Affiliation: Departamento de Matemáticas, Universidad del País Vasco, Aptdo. 644, 48080 Bilbao, Spain
Email: mtppealj@lg.ehu.es

DOI: http://dx.doi.org/10.1090/S0025-5718-03-01547-3
PII: S 0025-5718(03)01547-3
Received by editor(s): November 28, 2000
Received by editor(s) in revised form: July 5, 2002
Published electronically: May 30, 2003
Additional Notes: The second and third authors were supported by a grant from the University of the Basque Country.
Article copyright: © Copyright 2003 American Mathematical Society