On elliptic curves y2=x3n2x with rank zero

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Abstract

In this paper we determine all elliptic curves En:y2=x3n2x with the smallest 2-Selmer groups Sn=Sel2(En(Q))={1} and Sn′=Sel2(En′(Q))={±1,±n}(En′:y2=x3+4n2x) based on the 2-descent method. The values of n for such curves En are described in terms of graph-theory language. It is well known that the rank of the group En(Q) for such curves En is zero, the order of its Tate-Shafarevich group is odd, and such integers n are non-congruent numbers.

Keywords

Elliptic curve
Rank
Selmer group
2-descent method
Odd graph

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