j-Invariant

The -invariant is an invariant of an elliptic curve under isomorphism over an algebraic closure. For a general Weierstrass equation

(1)

define

			(2)
			(3)
			(4)
			(5)

The standard invariants of the equation are

			(6)
			(7)

Here, is the weight-four invariant and is the elliptic discriminant. For , the -invariant is

(8)

Over a field whose field characteristic is different from 2 and 3, an elliptic curve can be written in short Weierstrass form , in which case

(9)

Two elliptic curves over an algebraically closed field are isomorphic if and only if they have the same -invariant. Over a non-algebraically closed base field, curves with the same -invariant need not be isomorphic. Such curves may be twists of one another. Over a field whose field characteristic is different from 2 and 3, the exceptional values and correspond to curves having additional automorphisms.

Over , an elliptic curve is analytically isomorphic to a complex torus for in the upper half-plane, and its -invariant is the value of the modular j-function. The Fourier expansion and special values of , together with its role in the theory of complex multiplication, provide further arithmetic information about the -invariant.