Tamagawa number (reviewed)

The Tamagawa number of an elliptic curve \(E\) defined over a number field at a prime \(\mathfrak{p}\) of \(K\) is the index \([E(K_{\mathfrak{p}}):E^0(K_{\mathfrak{p}})]\), where \(K_{\mathfrak{p}}\) is the completion of \(K\) at \(\mathfrak{p}\) and \(E^0(K_{\mathfrak{p}})\) is the subgroup of \(E(K_{\mathfrak{p}})\) consisting of all points whose reduction modulo \(\mathfrak{p}\) is smooth.

The Tamagawa number of \(E\) at \(\mathfrak{p}\) is usually denoted \(c_{\mathfrak{p}}(E)\). It is a positive integer, and equal to \(1\) if \(E\) has good reduction at \(\mathfrak{p}\) and may be computed in general using Tate's algorithm.

The product of the Tamagawa numbers over all primes is a positive integer known as the Tamagawa product.