Global minimal model (reviewed)

A global minimal model for an elliptic curve \(E\) defined over a number field \(K\) is a Weierstrass equation for \(E\) which is integral and is a local minimal model at all primes of \(K\).

When $K$ has class number $1$ all elliptic curves over $K$ have global minimal models. In general, there is an obstruction to the existence of a global minimal model for each elliptic curve $E$ defined over $K$, which is an ideal class of $K$. In case this class is nontrivial for $E$, there is a semi-global minimal model for $E$, which is minimal at all primes except one, the ideal class of that one prime being the obstruction class.