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Yes, this is published as appendix A to chapter 3 in Derickx' PhD thesis available here: https://openaccess.leidenuniv.nl/handle/1887/43186 . The thesis contains, of course, many more interesting resu …

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^ …

Well spotted. This is a problem. After quite a bit of fiddling I found that the error is in Denis Simon's script used by Sage. In fact, when executed with higher values of the parameters so that the s …

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 …

Question 3: Yes, for plenty of curves one can calculate the 3-Selmer group and show that it is equal to the contribution from the Mordell-Weil group and hence that $Ш(E/\mathbb{Q})[3^{\infty}]$ is tri …

I fear you wish for too much here. If $Ш$ is finite, then we can represent each element by a torsor; each torsor has good reduction away from a finite set and the union of all bad places would then be …

Often it is, but not always. For instance if $K=\mathbb{Q}$ then the map is injective if and only if the rank of $E(\mathbb{Q})$ is at most $1$. This is because the $p$-adic elliptic logarithm of a no …

When there is a $2$-torsion point present then one should indeed use these isogenies to do a descent. One can do this over a number field as long as it is not too difficult to calculate the class grou …

The Kodaira symbol characterises the geometry of the special fibre, that is the structure over the algebraically closed field $\bar{\mathbb{F}}_p$. Good reduction means type I${}_0$, multiplicative re …

There is the Stickelberg element $\Theta$ considered by Mazur and Tate which gives more information in this direction. It is conjectured to be in the Fitting ideal and hence in the annihilator of the …

Let us assume that $E$ has complex multiplication by a maximal order $\mathcal{O}$. Then $E[\ell]$ is a free rank $1$ module over $\mathcal{O}/\ell\mathcal{O}$. The image $G$ of $\rho_{E,\ell}$ is con …

Given that you have not seen cyclotomic units, I think you should start with them. Rubin's appendix to Lang's book(s) on cyclotomic fields is one place or the book by Coates and Sujatha. Then for elli …

In other words, given an elliptic curve $E/\mathbb{Q}_p$ with additive reduction, you wish to know whether there is a quadratic extension $F/\mathbb{Q}_p$ such that $E/F$ has good reduction. Of course …

The comments above give already the answer, but for the sake of completeness let us be a bit more precise. Let $E/\mathbb{Q}$ be an elliptic curve. Let $\Omega^{+}$ be the smallest positive element i …

The remark added to the question shows that the kernel of $Ш(E/F) \to Ш(E/L)^G$ is finite where $G$ is the finite Galois group of $L/F$. $\DeclareMathOperator{\coker}{coker}$ Here is an argument why t …