Number of Points on Zp Curve for y^2 = x^3 + x^2

[Markjdb]

Let p be a prime number. Let Zp denote the field of integers modulo p. Determine the
number of points (x, y) with x, y in Zp such that y^2 = x^3 + x^2.

I just don't really have any idea how to approach this; the last problem was to find all rational points on the above curve, which I did, but I'm not quite sure where to start with this one.

 

[HallsofIvy]

An obvious way to start is to look at simple examples. If p= 2, then Zp consists only of 0 and 1 so the only possible points are (0,0), (1, 0), (0,1), and (1,1). How many of those satisfy the equation? If p= 3, Zp= Z3 consists of 0, 1, and 2 so there are 9 possible points. How many of them satisfy the equation?