Rank computations for the congruent number elliptic curves.
Rogers, Nicholas F. (2000)
Experimental Mathematics
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Displaying similar documents to “Mordell-Weil ranks of families of elliptic curves associated to Pythagorean triples”
Rank computations for the congruent number elliptic curves.
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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. ...