I've been reading about the elliptic curve used for key generation in bitcoin ie y^2 = x^3 + 7 (mod p) [I'm not sure how to do the congruence symbol]
To help me visualise whats going on I wrote a script to plot the values of y * y - (x * x * x + 7) (mod p) as colours (black = 0).
This is the result where p = 5
The plot is NOT symmetrical about y=p/2 but when I mark the integer values of x and y (the red dots) which yield a zero solution the result IS symmetrical (for all values of p)
Is there a neat mathematical reason for this?
First view the curve over $ℝ$. The equation $$E: Y^2 = X^3 + 7$$ has solutions clearly symmetrical about the horizontal axis $Y = 0$ because any real point $(x,y)$ solves the equation if and only if $(x,-y)$ does.
The very same is happening modulo $n$ for any integer $n$. The difference is that modulo $n$, you have $-y \equiv n-y$ for all integers $y$, which gives you another axis of symmetry. For every integer point $(x,y)$ and modulo $n$ $$(x,y) ∈ E ⇔ (x,-y) ∈ E ⇔ (x,n-y) ∈ E.$$
This now looks like a symmetry about the line $Y = \dfrac n 2$.
For example, for $n = 5$, you get the symmetries \begin{align*}(3,2) ∈ E ⇔ (3,-2) ∈ E ⇔ (3,3) ∈ E,\\ (4,1) ∈ E ⇔ (4,-1) ∈ E ⇔ (4,4) ∈ E. \end{align*}