j-invariant of an elliptic curve

Summary

The $j$-invariant of an elliptic curve is a number computed from the coefficients of a Weierstrass equation of the curve. It does not change under changes of coordinates. Over an algebraically closed field, two elliptic curves are isomorphic exactly when they have the same $j$-invariant. So the $j$-invariant is a coordinate on the set of isomorphism classes of elliptic curves. Over $\C$ it agrees with the value of the modular j-invariant at the period ratio $\tau$ of the curve.