Let $B\in K^\ast$, where $K$ is a number field. Let $y^2=X^3+B$ be the Weierstrass equation for an elliptic curve $E_B$ over $K$.
Note that the $j$-invariant of $E$ is zero.
When is $E_B$ isomorphic to $E_{B^\prime}$ over $K$? (Here $B^\prime \in K^\ast$.) Does this happen if and only if $B=B^\prime$? Or does it also happen if $B^\prime = u B$, where $u$ is a unit in $O_K^\ast$?
How do I determine the semi-stable reduction of $E_B$? That is, we know that $E_B$ has potential good reduction. How do I determine $L/K$ such that the elliptic curve $E_{B}\otimes_K L$ has good reduction over $O_L$?
If $B$ is a unit, then $E_B$ has good reduction. So we may and do assume $B$ is not a unit, i.e., $B$ is a prime.
I have a feeling that we must choose $L$ such that its ramification over the primes dividing $B$ is of the "right" type. But how do I see what the necessary type is?