Elliptic Curves
Complex Multiplication and Class Numbers
The link between curves with complex multiplication and class groups of imaginary quadratic orders, and the Hilbert class polynomial.
Engineering / MathematicsElliptic Curves2 min readKV-MATH-0639
Curves with complex multiplication by an imaginary quadratic order are in bijection with the ideal classes of that order. This is one of the most striking links in the subject and it is directly computational.
The correspondence
For a fixed imaginary quadratic order, the curves with that endomorphism ring correspond to the ideal classes. The number of such curves, up to isomorphism, is the class number.
The Hilbert class polynomial
The j-invariants of these curves are algebraic integers, and they are the roots of a single polynomial with integer coefficients — the Hilbert class polynomial of the discriminant.
| Property | Value |
|---|---|
| Degree | The class number of the order |
| Coefficients | Rational integers, and very large |
| Splitting field | The Hilbert class field of the imaginary quadratic field |
| Roots | The j-invariants of the CM curves |
Computing it
Computing the Hilbert class polynomial
- Enumerate reduced formsOf the given discriminant — see form reduction.
- Compute j-invariantsNumerically from each form, via the corresponding lattice.
- Form the productMultiply the linear factors numerically.
- RoundThe coefficients are integers; round and verify.
Alternative class invariants
Weber functions and other modular functions give class invariants whose minimal polynomials have substantially smaller coefficients while generating the same field. Converting back to j-invariants is a simple algebraic step, so these are always preferred in practice.
The application
Constructing a curve with known group order over a finite field is done by choosing a discriminant, computing the class polynomial, and finding a root modulo the prime. This is the engine of ECPP.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 7.2.4. Structural reference unverified: the source file was not available during authoring; chapter and section numbers are taken from the published edition and have not been checked against a physical copy.