Tamagawa numbers

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.