order finite field

edit

You can get the source code of the order method by typing:

sage: E.order??

As you can see, this is basically a handler that decides which algorithm fits best since you did not specify one. In your case, it will use pari, so you have to look at the pari source code.

You can also have a look at

sage: E.cardinality_bsgs??

which implements "Mestre's trick", or

sage: E.cardinality_exhaustive??

which counts vertically over each x-value.