Glasnik Matematicki, Vol. 48, No. 1 (2013), 49-58.

SUMS OF BIQUADRATES AND ELLIPTIC CURVES

Julián Aguirre and Juan Carlos Peral

Departamento de Matemáticas, Universidad del País Vasco UPV/EHU, Aptdo. 644, 48080 Bilbao, Spain
e-mail: julian.aguirre@ehu.es

e-mail: juancarlos.peral@ehu.es


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.

2010 Mathematics Subject Classification.   11G05.

Key words and phrases.   Elliptic curve, rank, biquadrate.


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DOI: 10.3336/gm.48.1.04


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