Integral points on elliptic curves defined by simplest cubic fields.
Duquesne, Sylvain (2001)
Experimental Mathematics
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Duquesne, Sylvain (2001)
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Stroeker, Roel J., de Weger, Benjamin M.M. (1994)
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P. G. Walsh (2009)
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Stroeker, Roel J., Tzanakis, Nikos (1999)
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Masanari Kida (2001)
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We prove that the -invariant of an elliptic curve defined over an imaginary quadratic number field having good reduction everywhere satisfies certain Diophantine equations under some hypothesis on the arithmetic of the quadratic field. By solving the Diophantine equations explicitly in the rings of quadratic integers, we show the non-existence of such elliptic curve for certain imaginary quadratic fields. This extends the results due to Setzer and Stroeker.
Roelof J. Stroeker, Benjamin M. M. de Weger (1999)
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de Weger, Benjamin M.M. (1998)
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Takaaki Kagawa (2001)
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Dongho Byeon (2004)
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Farzali Izadi, Foad Khoshnam, Arman Shamsi Zargar (2016)
Colloquium Mathematicae
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We construct a family of elliptic curves with six parameters, arising from a system of Diophantine equations, whose rank is at least five. To do so, we use the Brahmagupta formula for the area of cyclic quadrilaterals (p³,q³,r³,s³) not necessarily representing genuine geometric objects. It turns out that, as parameters of the curves, the integers p,q,r,s along with the extra integers u,v satisfy u⁶+v⁶+p⁶+q⁶ = 2(r⁶+s⁶), uv = pq, which, by previous work, has infinitely many integer solutions. ...
Rubin, Karl, Silverberg, Alice (2000)
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Clemens Fuchs, Rafael von Känel, Gisbert Wüstholz (2011)
Acta Arithmetica
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Sungkon Chang (2010)
Acta Arithmetica
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