Displaying similar documents to “On a characterization of Shimura's elliptic curve over ℚ(√37)”

Good reduction of elliptic curves over imaginary quadratic fields

Masanari Kida (2001)

Journal de théorie des nombres de Bordeaux

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We prove that the j -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.

Rank of elliptic curves associated to Brahmagupta quadrilaterals

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. ...