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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.

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• Last edited by Andrew Sutherland on 2019-03-09 15:02:36
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