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

Question

I need to prove that the rank of the curve $y^2=x^3+px$ is $0$, if $p\equiv 7 \pmod {16}$ is a prime. Using the standard technique, we need to show that none of the following two equations admits an integer solution in M, N and e (with M, N and e pairwise coprime; M, e non-zero):
$2M^4-2pe^4=N^2$
$4M^4-pe^4=N^2$

I have got this after going modulo 16. But, $2M^4-2pe^4=N^2$ and $4M^4-pe^4=N^2$ do admit solutions in $Z/16Z$

Answer 1

This is shown in Silverman's "The Arithmetic of Elliptic Curves", Chapter X, Section 6 (The curve $Y^2=X^3+DX$), Proposition 6.2.