Let be a field over which the curve is defined (that is, the coefficients of the defining equation or equations of the curve are in ) and denote the curve by . Then the -rational points of are the points on whose coordinates all lie in , including the point at infinity. The set of -rational points is denoted by . is a group, because properties of polynomial equations show that if is in , then is also in , and if two of , , are in , then so is the third. Additionally, if is a subfield of , then is a subgroup of .
Let be a field over which the curve is defined (that is, the coefficients of the defining equation or equations of the curve are in ) and denote the curve by  Then the -rational points of are the points on whose coordinates all lie in , including the point at infinity. The set of -rational points is denoted by . is a group, because properties of polynomial equations show that if is in , then is also in , and if two of , , are in , then so is the third. Additionally, if is a subfield of , then is a subgroup of .