Tate's algorithm

In the theory of elliptic curves, Tate's algorithm takes as input an integral model of an elliptic curve E over ${\displaystyle \mathbb {Q} }$, or more generally an algebraic number field, and a prime or prime ideal p. It returns the exponent f_p of p in the conductor of E, the type of reduction at p, the local index

where ${\displaystyle E^{0}(\mathbb {Q} _{p})}$ is the group of ${\displaystyle \mathbb {Q} _{p}}$-points whose reduction mod p is a non-singular point. Also, the algorithm determines whether or not the given integral model is minimal at p, and, if not, returns an integral model with integral coefficients for which the valuation at p of the discriminant is minimal.

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