Tate's algorithm determines the local reduction data of an elliptic curve at a finite place. Let 
be a local field with discrete
valuation 
,
valuation ring 
, and residue field 
. The input is an integral equation in Weierstrass
form
 |
with 
and nonzero elliptic discriminant. Through
admissible changes of variables and tests in 
, the algorithm finds a local minimal equation and analyzes
its reduction.
The output includes the reduction type (good, multiplicative, or additive), the Kodaira-Néron symbol, the valuation 
of the minimal elliptic discriminant, the
local conductor exponent 
, and the local Tamagawa number. The possible Kodaira-Néron
symbols are 
,
, 
, 
, 
, 
, 
, 
, and 
. If 
is the number of irreducible components of the special fiber
of the Néron minimal model, then Ogg's formula gives
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At residue characteristics greater than 3, 
is 0, 1, or 2 according as the reduction is good, multiplicative,
or additive. At residue characteristics 2 and 3, the algorithm also accounts for
wild ramification. Applying it at the prime
ideals of a number field determines the local
factors of the elliptic curve conductor.
