Tate's algorithm

In the theory of elliptic curves, Tate's algorithm takes as input an integral model of an elliptic curve $E$ over $\mathbb {Q}$ and a prime $p$. It returns the exponent $f_{p}$ of $p$ in the conductor (mathematics) of $E$, the type of reduction at $p$, the local index

$c_{p}=[E(\mathbb {Q} _{p}):E^{0}(\mathbb {Q} _{p})]$,

where $E^{0}(\mathbb {Q} _{p})$ is the group of $\mathbb {Q} _{p}$-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 which is minimal at $p$.

The type of reduction is given by the Kodaira symbol, for which, see elliptic surfaces.

References

Cremona, John. "Algorithms for modular elliptic curves". Retrieved 2007-12-20.