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The Tamagawa number of an elliptic curve EE defined over a number field at a prime p\mathfrak{p} of KK is the index [E(Kp):E0(Kp)][E(K_{\mathfrak{p}}):E^0(K_{\mathfrak{p}})], where KpK_{\mathfrak{p}} is the completion of KK at p\mathfrak{p} and E0(Kp)E^0(K_{\mathfrak{p}}) is the subgroup of E(Kp)E(K_{\mathfrak{p}}) consisting of all points whose reduction modulo p\mathfrak{p} is smooth.

The Tamagawa number of EE at p\mathfrak{p} is usually denoted cp(E)c_{\mathfrak{p}}(E). It is a positive integer, and equal to 11 if EE has good reduction at p\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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