Miyake, Katsuya Some families of Mordell curves associated to cubic fields. (English) Zbl 1080.14520 J. Comput. Appl. Math. 160, No. 1-2, 217-231 (2003). Summary: We introduce some Mordell curves of two different natures both of which are associated to cubic fields. One set of them consists of those elliptic curves whose rational points over the rational number field are described by or closely related to cubic fields. The other is a one-parameter family of Mordell curves which gives all (cyclic) cubic twists and all quadratic twists of the Fermat curve \(X^3+Y^3+Z^3=0\). Cited in 2 ReviewsCited in 2 Documents MSC: 11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields 11G05 Elliptic curves over global fields 14H25 Arithmetic ground fields for curves PDFBibTeX XMLCite \textit{K. Miyake}, J. Comput. Appl. Math. 160, No. 1--2, 217--231 (2003; Zbl 1080.14520) Full Text: DOI References: [1] Birch, B. J.; Swinnerton-Dyer, H. P.F., Notes on elliptic curves I, II, J. Reine Angew. Math., 218, 79-108 (1965) · Zbl 0147.02506 [2] Breuil, C.; Conrad, B.; Diamond, F.; Taylor, R., On the modularity of elliptic curves over Qwild 3-adic exercises, J. Amer. Math. Soc., 14, 4, 843-939 (2001), (electronic) · Zbl 0982.11033 [3] Cassels, J. W.S., The rational solutions of the diophantine equation, \(y^2=x^3\)−\(D\), Acta Math., 82, 243-273 (1950) · Zbl 0037.02701 [4] Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpern (German), Invent. Math., 73, 349-366 (1983), (Erratum. Invent. Math. 75 (1984) 381) · Zbl 0588.14026 [5] Fueter, R., Ueber kubische diophantische Gleichungen, Comm. Math. Helv., 2, 69-89 (1930) · JFM 56.0877.01 [6] Gebel, J.; Pethö, A.; Zimmer, H. G., On Mordell’s equation, Comp. Math., 110, 335-367 (1998) · Zbl 0899.11013 [7] Mestre, J.-F., Rang de coubres elliptiques d’invariant donné, C. R. Acad. Sci. Paris, Sér. I, 314, 919-922 (1992) · Zbl 0766.14023 [8] Mestre, J.-F., Rang de coubres elliptiques d’invariant nul, C. R. Acad. Sci. Paris, Sér. I, 321, 1235-1236 (1995) · Zbl 0853.11047 [9] Mordell, L. J., The Diophantine equation \(y^2\)−\(k = x^3\), Proc. London Math. Soc., 13, 60-80 (1914) · JFM 44.0230.03 [10] Mordell, L. J., Diophantine Equations (1969), Academic Press: Academic Press London, New York · Zbl 0188.34503 [11] Nagel, T., Ueber die rational Punkte auf einige kubischen Kurven, Tôhoku Math. J., 24, 48-53 (1924) · JFM 50.0089.03 [12] Selmer, E. S., Completion of the table, Acta Math., 92, 191-197 (1954) · Zbl 0056.26704 [13] Silverman, J. H., The Arithmetic of Elliptic Curves (1986), Springer: Springer New York · Zbl 0585.14026 [14] Silverman, J. H.; Tate, J., Rational Points on Elliptic Curves (1992), Springer: Springer New York · Zbl 0752.14034 [15] Wiles, A., Modular elliptic curves and Fermat’s last theorem, Ann. of Math., 141, 443-551 (1995) · Zbl 0823.11029 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.