I was reading this tutorial about elliptic curves.
There are some example curves with $p=19$, $97$, $127$, $487$ etc, and they are all symmetric about $y=p/2$. Why is it the case?
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3The plot is symmetrical because if you write the curve as $y^2 = x^3 + ax + b$ then the solutions have a symmetry $(x,y) \leftrightarrow (x,-y)$, which reduces mod $p$ gives $(x,y) \leftrightarrow (x,p-y)$ i.e. reflection above $y = p/2$. There can be an exception when $y=0$, but then the number of points mod $p$ is even, which for many applications is undesirable (we want a the number of points mod $p$ to be prime). – Noam D. Elkies Jan 10 '23 at 01:37
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@NoamD.Elkies Thanks, but why when you reduce it modulo p, the symmetry becomes (x,y)↔(x,p-y)? – frt132 Jan 10 '23 at 01:53
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Not sure what kind of explanation you want. That's what -y becomes mod p if 0 < y < p. Note that p is 0 mod p so p-y is -y mod p. – Noam D. Elkies Jan 10 '23 at 02:36
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@NoamD.Elkies I see, thanks very much. – frt132 Jan 10 '23 at 05:22