in Silverman Tate, Rational Points on Elliptic Curves the exercise 4.6 describes a special case of a theorem of Eichler and Shimura, the exercise is about the elliptic curve $$C:y^2=x^3-4x^2+16,$$ let $M_p=\#C(\mathbb{F}_p)$ and $F(q)$ the formal power series $$ F(q)=q\prod_{n=1}^{\infty}(1-q^n)^2(1-q^{11n})^2=q-2q^2-q^3+2q^4+\dotsb $$ let $N_m$ be the coefficient of $q^n$ in $F(q)$ $$F(q)=\sum_{n=1}^{\infty}N_mq^n$$.
The exercise ask for compute the sum $M_p+N_p$ and formulate and prove a conjecture.
I run a computation in Sage and its seems that the conjecture is that for every prime $p$ we have $M_p+N_p=p$. I have the following questions
Is this the result of Eichler and Shimura?
I see the product in LMFB and $F(q)$ is equal to $\eta(z)^2\eta(11z)^2$, so I think that the curve $C$ is the modular curve $X_0(11)$, is this correct?
I saw the same product in Taylor Modular Arithmetic: Driven by Inherent Beauty and Human Curiosity, in this case for the curve $y^2+y=x^3-x^2$, are this the same curve $C$? (I mean birrationaly isomorphic curves)
I dont know about modular forms, I look at the paper The modular curves $X_0(11)$ and $X_1(11)$, and its seems that there is a path from the modular form to the elliptic curve, there is also a path that begins in the elliptic curve and produces $F(Q)$? (I know that this is the Modularity Theorem Shimura-Taniyama), I mean in this specific case there is a elementary path that produces the modular form $F(q)$ from the curve $C$?
In general what are the suggestions for this special case, I mean in this case is an exercise of a book for undergraduates, so I think that the author considers that an advanced undergraduate can do this exercise, but I don see any suggestions in the book.
Thanks in advance! Sorry for my bad English .