Experimental Mathematics

Rank computations for the congruent number elliptic curves

Nicholas F. Rogers

Abstract

In a companion paper, Rubin and Silverberg relate the question of unboundedness of rank in families of quadratic twists of elliptic curves to the convergence or divergence of certain series. Here we give a computational application of their ideas on counting the rational points in such families; namely, to find curves of high rank in families of quadratic twists. We also observe that the algorithm seems to find as many curves of positive even rank as it does curves of odd rank. Results are given in the case of the congruent number elliptic curves, which are the quadratic twists of the curve $y^2 = x^3 - x$; for this family, the highest rank found is 6.

Article information

Source
Experiment. Math., Volume 9, Issue 4 (2000), 591-594.

Dates
First available in Project Euclid: 20 February 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1045759524

Mathematical Reviews number (MathSciNet)
MR1806294

Zentralblatt MATH identifier
1050.11061

Subjects
Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 11G50: Heights [See also 14G40, 37P30]

Citation

Rogers, Nicholas F. Rank computations for the congruent number elliptic curves. Experiment. Math. 9 (2000), no. 4, 591--594. https://projecteuclid.org/euclid.em/1045759524


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