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Rank computations for the congruent number elliptic curves. (English) Zbl 1050.11061

Summary: In a companion paper, K. Rubin and A. Silverberg [Exp. Math. 9, No. 4, 583–590 (2000; Zbl 0959.11023)] 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.

MSC:

11G05 Elliptic curves over global fields
11G50 Heights

Citations:

Zbl 0959.11023

Software:

APECS; mwrank
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References:

[1] Connell I., ”APECS: Arithmetic of Plane Elliptic Curves”
[2] Cremona J. E., ”mwrank, a program for 2–descent on elliptic curves over Q” (1998)
[3] DOI: 10.2307/2939253 · Zbl 0725.11027
[4] DOI: 10.1007/978-1-4684-0255-1
[5] DOI: 10.1007/BF01451411 · Zbl 0561.10007
[6] DOI: 10.3792/pjaa.74.29 · Zbl 0922.11023
[7] DOI: 10.3792/pjaa.69.175 · Zbl 0795.11012
[8] Rubin K., Experiment. Math. 9 (4) pp 583– (2000)
[9] Silverman J. H., The arithmetic of elliptic curves 106 (1986) · Zbl 0585.14026
[10] DOI: 10.1007/BF01389327 · Zbl 0515.10013
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