I am currently reading a paper by De Weger and one theorem in it proves a bound for the Tamagawa number of any elliptic curve defined over $\mathbb{Q}$.

I was wondering if anyone has any good references/texts that provide an exposition on the Tamagawa number of an elliptic curve as I was unable to find one in the Arithmetic of Elliptic Curves.

I know the definition of the Tamagawa number from this reference, but not really much more than that (http://math.uci.edu/~asilverb/connectionstalk.pdf).

EDIT: I am looking for something a little more in depth than the intuition behind the definition. For instance useful recent applications of Tamagawa numbers or what is known about Tamagawa numbers for elliptic curves over $\mathbb{Q}$.