  Let $K$ be a global field and let $E/K$ be an elliptic curve with a
$K$-rational point of prime order $p$. In this paper we are interested in how
often the (global) Tamagawa number $c(E/K)$ of $E/K$ is divisible by $p$. This
is a natural question to consider in view of the fact that the fraction
$c(E/K)/ |E(K)_{\text{tors}}|$ appears in the second part of the Birch and
Swinnerton-Dyer Conjecture. We focus on elliptic curves defined over global
fields, but we also prove a result for higher dimensional abelian varieties
defined over $\mathbb{Q}$.
