i have read many books concerning the definition of tamagawa numbers ,but none of the books explained an intuition behind the concept , i mean what could be the intuitive definition of tamagawa number

i am expecting some other explanation ,other than the ones present in the textbooks,

i wanted to know the reason why the "prof.Peter swinnerton dyer " used the tamagawa number as a central part in conjecturing his work about elliptic curves, what is the reason behind that ???

what do we get when we calculate the tamagawa numbers of the elliptic curve ,i mean i want a logical imaginative definition of it,

hope you understood,