Simple Constructions for Elliptic Cubic Curves with Specified Small Numbers of Points

Abstract

In a finite desarguesian projective plane of order q n , the number of points which an elliptic cubic curve can contain is bounded above and below by the Hasse-Weil bounds q + 1 ± ⌊2√ q ⌋. Our purpose here is to provide very simple actual constructions for the equation of a curve which is required to have a specified small number of points. As a biproduct we show that an elliptic cubic curve having an odd multiple of 7 points cannot exist in a plane of order 2 r , r odd, and that such a curve having an odd multiple of 11 points cannot exist in a plane of order 2 r if r is relatively prime to 5.