Difference between exclamation mark (!) and point (.) in MAGMA

Question

In this MAGMA handbook, I found the following code:

K := FiniteField(2, 160); // finite field of size 2^160
E := EllipticCurve([K!1, 0, 0, 0, K.1]);

There, I noted the notations K!1 and K.1, and I do not really understand what the difference is between these two. It seems like two ways of conversion of $1$ into the finite field $K$. But then it would be pretty useless to use both notations if they supposedly do the same. Could someone explain this to me?

Answer 1

The ! denotes coercion, for which more information can be found here: http://magma.maths.usyd.edu.au/magma/handbook/text/9

On the other hand, K.1 picks out the first generator of K.

Consider for instance the code:

F<x> := FiniteField(3,2); \\ F.1; \\ F!1;

With output:

x; \\ 1

So they are very much not the same (but may happen to overlap in simple cases)!