Rank of Elliptic Curve, $Y^2=x^3+px$ where $p$ is prime is either $0,1,2$

Question

I am following book "Rational Points on Elliptic Curves" by Silverman-Tate(basic version not the "The Arithmetic of Elliptic Curves" by Silverman-Tate) And I am trying to solve for cubic curve, $y^2=x^3+px$ where $p$ is prime

1) Rank of the curve is $0,1$ or$ 2$.

Using the standard methods described in that book. I am considering various equations of the form $N^2 = b1M^4+b2e^4$ where $b_1.b_2=b$ to check if there is any solutions modulo $p$ and considering quadratic residue modulo $p$.

I know that this same questions were asked previously but currently I am not familiar with the content of "The Arithmetic of Elliptic Curves" by Silverman-Tate)". So I need to prove what is currently in "Rational Points on Elliptic Curves" (by Silverman-Tate).

Any kind of help is much appreciated.

Answer 1

This is proved in Silverman's book "The Arithmetic of Elliptic Curves" on page 311. The rank $r(E)$ of $E:y^2=x^3+px$ for $p$ prime is given as follows: $$ r(E)= \begin{cases} 0, \text{ if } p\equiv 7,11 \bmod 16 \\ 0 \text{ or }1, \text{ if } p\equiv 3,5,13,15 \bmod 16 \\ 0 \text{ or }1 \text{ or }2, \text{ if } p\equiv 1,9 \bmod 16 \end{cases} $$ You can compare with the book by Silverman-Tate, in order to see which theorems are available.

Edit. After searching I found the question you have mentioned:

Rank of the elliptic curve $y^2=x^3+px$