Original scientific paper

https://doi.org/10.3336/gm.48.1.04

Sums of biquadrates and elliptic curves

Julián Aguirre ; Departamento de Matemáticas, Universidad del País Vasco UPV/EHU, Aptdo. 644, 48080 Bilbao, Spain
Juan Carlos Peral ; Departamento de Matemáticas, Universidad del País Vasco UPV/EHU, Aptdo. 644, 48080 Bilbao, Spain

Abstract

Given the family of elliptic curves y2= x3-(1+u4) x, uQ, or equivalently y2=x3-(m4+n4)x for m,n integers, we prove that its rank over Q(u) is 2. We also show the existence of subfamilies of rank at least 3 and 4 over Q(u). Also, assuming the Parity Conjecture, we prove the existence of infinitely many curves having rank at least 5 over Q.
Performing an exhaustive search in the range 1 ≤ n < m ≤ 251000 we have found more than 1500 curves with rank 8, over 150 with rank 9, nine of rank 10 and one of rank 11. This improves previous results of Izadi, Khoshnam and Nabardi.

Keywords

Elliptic curve; rank; biquadrate

Hrčak ID:

103269

URI

https://hrcak.srce.hr/103269

Publication date:

4.6.2013.

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