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Let $K$ be a finite extension of $\mathbb{Q}_p$ with $p\neq 2$. Suppose that $E/K$ is an elliptic curve with additive reduction and such that $E$ has full $4$-torsion over $K$. By the Kodaira classif …

Again an early source explaining how this is done in practice is Cremona's book. Specifically section 3.8. One current implementation for finding the isogeny class of an elliptic curves over a number …

(Not sure any of these questions are at the right level for this forum, but here the comments that may help.) question : Inflation-restriction sequence. question : The target can be identified with …

[Rather than leaving the comment "Class number formula" for Olivier as a comment, I expand it for other readers of the question, to a partial answer.] Kummer knew in 1850 that the class group of $\mat …

Let $Z$ be the number of zeroes of $y\mapsto f(iy)$ for $0<y<\infty$ and let $r$ be the analytic rank of $E$. Then $Z\geqslant r$ and $Z\equiv r \pmod{2}$ as I will explain below. I don't know of an e …

Let $\sigma$ be the non-trivial element of the Galois group of the quadratic extension $L/K$. Let $\phi \colon E \to E_D$ be the isomorphism defined over $L$. First, if $P \in E(\bar L)$ then $\sigma\ …

As in the question $K$ is a number field and $E/K$ an elliptic curve. Let me start by saying that I think the best analogues are the following two short exact sequences (the first two "$/n$" means mod …

I think it is a good idea to compare the growth of the rank to the degree. I would say that we have excessive growth in an extension $F/K$ if $$\DeclareMathOperator{\rank}{rank}\rank E(F) - \rank E(K) …

There is a formula but it involves both cubic twists. Let $E: y^2 = x^3+B$ be an elliptic curve over $\mathbb{Q}$ with $j=0$ as the one in the question. Let $D$ be a cubefree integer. Set $E_1: y^2=x^ …

As far as I understand the question (I know elliptic curves, but I don't know what MU and BP are), the task is to express the coefficients of the Weierstrass equation given the series of multiplicatio …

This is not an answer, but an idea how one might get an answer with quite a bit of work. I am not too confident that I have not overlooked something. Let $p$ be a prime and $k>1$. I am aiming to const …

(Not sure why this question comes up naturally. The more interesting question, and the one analogue to the kernel of restriction, is to ask what is the cokernel of corestriction. That turns up a lot. …

Let $S$ be a finite set containing all places where an elliptic curve $E$ has bad reduction as well as $p$ and $\infty$. Write $T$ for $T_pE$ and $G_S$ for the Galois group of the maximal extension o …

By cm theory, $E[p]$ is isomorphic to $\mathcal{O}_K/(p)$ as an $\mathcal{O}_K$-module. Elements in $\operatorname{Gal}(\bar{\mathbb{Q}}/K)$ act $\mathcal{O}_K$-linearly on $E[p]$ since they commute w …

Here a long comment to settle this question. This is really a bug in the implementation of $p$-adic heights in sagemath. I have announced it as a bug here on the sage trac list. I hope to add the code …