Associated with an elliptic curve E/K over a number field K is a finite set of integers greater than 1 called the local Tamagawa numbers of E/K. The ratio (product of the Tamagawa numbers)/|Torsion in E(K)| appears in the conjectural leading term of the L-function of E in the Birch and Swinnerton-Dyer conjecture, and we are interested in understanding whether there are cancellations in this ratio when E(K) has a non-trivial torsion subgroup.

When N is prime, let us call N-special an elliptic curve E/K with a K-rational torsion point of order N and such that N does not divide the product of the Tamagawa numbers. We will show that the existence of an N-special elliptic curve E/K is intimately linked to the existence of exceptional units in the ring of integers of K. When N > 2d+1, we suggest that there exist only finitely fields K/Q of degree d having (finitely many) N-special elliptic curves E/K. The list of known N-special elliptic curves is surprisingly short when d is at most 7.