Tate's algorithm [SEP] named after
characteristic of the residue class field is not 2 or 3; in this case the type and "c" and "f" can be read off from the valuations of "j" and Δ (defined below).
Tate's algorithm was introduced by as an improvement of the description of the Néron model of an elliptic curve by .
Notation.
Assume that all the coefficients of the equation of the curve lie in a complete discrete valuation ring "R" with perfect residue field and maximal ideal generated by
: this is the Néron–Ogg–Shafarevich criterion.
Ogg's formula expresses the conductor in terms of the discriminant and the number of components of the special fiber over a local field, which can be computed using Tate's algorithm.
History.
The conductor of an elliptic curve over a local field was implicitly studied (but not named) by in the form of an integer invariant ε+δ which later turned out to be the exponent of the conductor.
The conductor of an elliptic curve over the rationals was