# Graphically representing points on Elliptic Curve over finite field

I have taken elliptic curve $E\colon y^2=x^3-4x+20$, defined over $\mathbb{F}_{29}$. The number of points on the curve, $\left|E(\mathbb{F}_{29})\right|=37$.

I took base point $P=(1,5)$, and got following results: I want the results graphically to know the behaviour of the curve.

Is there any tool for this to show results graphically by giving input as curve points?

• Well, you could simply plot them as Cartesian pairs, but why do you want a graphical representation? Any way you try plotting them will loose information, since they are pairs in $\mathbb{F}_{29}\times\mathbb{F}_{29}$. – figlesquidge Dec 3 '13 at 11:29
• side question: How can you find a base point of a curve?Or because the underlying field is of prime order so all the points of the curve they do form a basis? – curious Dec 3 '13 at 17:20
• @curious: In this case it is simple because the curve has a prime number of points. – figlesquidge Dec 3 '13 at 17:30
• What does this mean? – curious Dec 3 '13 at 17:32
• @curious: Because the curve has a prime number of points, then (since the points on elliptic curves form groups) the curve must be isomorphic to $C_{37}$, which means any point (other than the point at infinity) is a generator. – figlesquidge Dec 10 '13 at 16:00

The point is an elliptic curve doesn't actually look anything like a curve when evaluated over a finite field, and indeed there's no real way of visualising points in $\mathbb{F}_{29}\times\mathbb{F}_{29}$ that doesn't loose some of the structure anyway.
The best I can think of would be to plot them simple as points in a $29\times29$ grid and join the dots, but even this would be painfully unclear, since for example when would you 'join' them by overflowing the grid (eg: would the line from $(1,5)\to(4,19)$ go straight from one to the other, or would it go out one side and reappear on the opposite side?).
If you were to evaluate $E(\mathbb{R})$ then the resulting points would form a curve in $\mathbb{R}^2$, which you could then plot.