I am working with a plane algebraic curve $C$ defined over the rational numbers $\mathbb{Q}$ that has singularities over an extension of $\mathbb{Q}$.
The problem is that the equation defining $C$ is pretty difficult and I am not sure about how to determine the extension of $\mathbb{Q}$ where the singularities are defined.
Could we use Magma computer algebra system to define $C$ over a field $K$ such that all the singularities of $C$ are defined over $K$?
The examples that I have found in Magma's handbook assume that the user knows which extension to move to.