Since the underlying field for the elliptic curve for secp256k1 is F_{p} where p=2^{256}-2^{32}-977, and since p is very close to a power of 2,

it makes sense to try to graph the elliptic curve y^{2}=x^{3}+7 over the field of 2-adic numbers or related rings. Since we only have a finite amount of room to post here, let us simply graph the elliptic curve over a ring of the form Z_{2n}.

The ring of 2-adic integers is the inverse limit of the rings of the form Z_{2n}. Therefore, the rings Z_{2n} can be thought of as finite approximations for the ring of 2-adic integers. Now, we want to graph the elliptic curve in such a way so that two points which are near each other with respect to the 2-adic metric are also near each other on the graphical representation of the elliptic curve.

Let f:Z_{2n}->{0,…,2^{n}-1} be the function where

f(a_{0}2^{0}+…+a_{n-1}2^{n-1})=

a_{n-1}2^{0}+…+a_{0}2^{n-1}

whenever a_{0},…,a_{n-1} are all elements of the set {0,1}. In other words, f simply reverses the bits in the binary representation of an element of

Z_{2n}. Then the white pixels in the following graph are precisely at the points with coordinates (f(x),f(y)) where y^{2}=x^{3}+7 mod 2^{n} where n=9. This picture is an approximation of the image of the elliptic curve over the ring of 2-adic integers.

Since the field of complex numbers is isomorphic to any ultraproduct of the algebraic closures of finite fields F_{p} by a non-principal ultrafilter on the set of all prime numbers, one should think of the finite fields as an approximation to the set of all complex numbers. Furthermore, the field of p-adic numbers embeds into the field of complex numbers, so one may think of the finite fields as objects that approximate a field that contains the field of p-adic numbers as a sub-field. Therefore, it is appropriate to use the graphs of the elliptic curve y^{2}=x^{3}+7 over the reals, complex numbers, or even the p-adic numbers as a visualization for the fields used in elliptic curve cryptography. One just needs to realize that one is working in a different field for visualization purposes.