It seems that most introductory books on elliptic curves simply state the definition of the j-invariant of an elliptic curve without giving any background on how that definition was conceived. Of course, for moduli reasons, it is clear why one might want such an invariant, but the actual formula has always seemed quite mysterious to me. Does anyone know of a nice self-contained source that explains the definition of the j-invariant?
Definition of the j-invariant of an elliptic curve
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1The complete definition, or just the definition up to a constant? Matt has given a nice explanation for why one should care about the j-invariant up to a constant, but the precise form of the j-invariant has to do with its number-theoretic properties and is not easy to explain in an introduction. On the other hand books on elliptic curves don't want to give you the wrong definition! So there is a dilemma here. – 2011-02-12