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Contents
1 Notation
2 The algorithm
3 Implementations

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 a prime π. The elliptic curve is given by the equation

y^2+a1xy+a3y=x^3+a2x^2+a4x+a6.
Define:
a{i,m}=a_{i}/π^m
b2=a1^2+4a2
b4=a1a3+2a4
b6=a3^2+4a6
b8=a1^2a6-a1a3a4+4a2a6+a2a3^2-a4^2
c4=b2^2-24b4
c6=-b2^3+36b2b4-216b6
Δ =-b2^2b8-8b4^3-27b6^2+9b2b4b6
j=c4^3/Δ .

Implementations
The algorithm is implemented for algebraic number fields in the PARI/GP computer algebra system, available through the function elllocalred.
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