For a specific example, you can do much better in practice. Look at Nigel Smart's "The algorithmic resolution of diophantine equations." Chapter VII.4, or other suggestions given in these answers

Serre in his Bourbaki talk gave a different proof than in Washington.

Theorem 13.12 is precisely this theorem when $\mathcal{O}=\mathbb{Z}_p$. What is your question?

@GHfromMO But there are three points at infinity for this curve. I know you are trying to take the proof of what the Jacobian of the projective closure is and get the minimal part of it to show that the ring is non PID. But I believe you need to be more careful here.

What is $O$ here?

The formulae are spelled out in Silverman's "Arithmetic of elliptic curves" chapter IV. They are implemented in many place, for instance in sage do: sage: R.<a,b> = QQ[]; K = R.fraction_field(); E = EllipticCurve(K,[a,b]); Ehat = E.formal_group(); Ehat.mult_by_n(2, prec = 30) to get a series that starts like $2\,t - 12\,a\,t^5 - 54\,b\,t^7 + 44\,a^2\,t^9$. Not sure if that is what you are looking for.

For $E$ without rational $2$-torsion, I would look for examples among those curves for which the $2$-torsion of the class number of $\mathbb{Q}(E[2])$ is large.

@DavidMcKinnon The universal elliptic curve over $X_1(p^k)$ is precisely such a family. For each $K$ it has finitely many $K$-rational fibres. But how would you show that there is one of rank $0$?

I agree, if $E$ is a curve that is not coming from a lower field, having a $n$-torsion point, I bet chances are good that the rank is $0$ or $1$. On the other hand for certain Galois groups the parity results coming from Brauer relations impose rank growth. I have not thought about $n=64$. But in any case, all I can offer so far is speculation that this should exist, but no idea how to construct an example or how to verify it.

One could state the main conjecture by saying that the fractional ideals in $\mathbb{Q}_p[\![T]\!]$ are equal and then note that the analytic side is integral and hence so is the algebraic side. However, if you read on you will see that they later state things without $\otimes\mathbb{Q}_p$ and $h$ is really the characteristic series of a $\Lambda$-module; it is just in the introduction that they aimed to simplfy the exposition. (... and that is my last comment on this page).

$(\gamma-1)(v) = \gamma(v) - v$ as always when you translate an action of a group $G$ on an $R$-module into a $R[G]$-module.

$T$ lies in the Iwasawa algebra, which is the projective limit of the group rings $\mathbb{Z}_p[G_n]$ with $G_n\cong (1+p\mathbb{Z}_p)/(1+p^{n+1}\mathbb{Z})$. As $\gamma$ maps to an element in $G_n$, the group ring element $T = \gamma-1$ is not in $G_n$, it is the formal difference of two group elements. All of this can be learned from a basic text on Iwasawa theory. Lang's or Washington's Cyclotomic Fields for instance.

The wikipedia article could certainly be improved, but $T=\gamma-1$ is used in both to identify $\Lambda = \mathbb{Z}_p[\![\Gamma]\!]$ with $\mathbb{Z}_p[\![T]\!]$.

To make 2 non-trivial you may want to replace general by simple. The arxiv link for the above comment answers the more interesting question, I believe.

I am not sure why you keep updating this. Max and my comment solved the initial question. I doubt many people will want to follow your coding experiments without extra motivation or explanations. (But if there is, maybe a new question is better than adding to this).

I believe the modularity of this elliptic curve is well explain in these notes. The curve in question here is isomorphic to $y^2+y=x^3-x$, which is shown to be the equation of $X_1(11)$. You will see this is really a bit more difficult than the level of Silverman-Tate, yet the notes give a good introduction into the subject.

Very well formulated and well referenced question.