Example
An example of an elliptic curve with complex multiplication is
- C/Zθ
where Z is the Gaussian integer ring, and θ is any non-zero complex number. Any such complex torus has the Gaussian integers as endomorphism ring. It is known that the corresponding curves can all be written as
- Y2 = 4X3 − aX,
having an order 4 automorphism sending
- Y → −iY, X → −X
in line with the action of i on the Weierstrass elliptic functions.
This is a typical example of an elliptic curve with complex multiplication. Over the complex numbers, all elliptic curves with complex multiplcation can be similarly constructed. That is, as quotients of the complex plane by some order in the ring of integers in an imaginary quadratic field.
Read more about this topic: Complex Multiplication
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