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Class Polynomials








Class polynomials are invariants of elliptic curves with complex 
multiplication by an imaginary quadratic order of discriminant D.  
As such the Hilbert class polynomials can be interpreted as defining 
a subscheme or divisor on the modular curve X(1) isomorphic to P^1, while 
the Weber variants define a subscheme of a modular curve of higher 
level.



HilbertClassPolynomial(D) : RngIntElt -> RngUPolElt



Given a negative discriminant D, returns the Hilbert class polynomial, 
defined as the minimal polynomial of j(tau), where Z[tau] is an 
imaginary quadratic order of discriminant D.



WeberClassPolynomial(D) : RngIntElt -> RngUPolElt



Given a negative discriminant D congruent to 1 modulo 8, 
returns the Weber class polynomial, defined as the minimal polynomial 
of f(tau), where Z[tau] is an imaginary quadratic order of 
discriminant D and f is a particular normalized Weber function
generating the same class field as j(tau).  A root f(tau) of 
the Weber class polynomial is an integral unit generating the ring 
class field related to the corresponding root j(tau) of the 
Hilbert class polynomial by the expression 


j(tau) = ((f(tau)^(24) - 16)^3 /f(tau)^(24)),


where ( GCD)(D, 3) = 1, and 


j(tau) = ((f(tau)^8 - 16)^3 /f(tau)^8),


if 3 divides D.  For further details, consult Yui and 
Zagier [YZ97].





Example CrvMod_Standard class polynomials (H93E5)

Class polynomials are typically used for constructing elliptic curves 
with a known endomorphism ring or known number of points over some 
finite field.  The Weber (and other) variants of the class polynomials 
were introduced as a means of obtaining class invariants -- defining 
the j-invariant of curves with given CM discriminant -- with much 
smaller coefficients.  In this example we give the classical example 
of D = - 71, where the 





> HilbertClassPolynomial(-71);
x^7 + 313645809715*x^6 - 3091990138604570*x^5 + 98394038810047812049302*x^4 
- 823534263439730779968091389*x^3 + 5138800366453976780323726329446*x^2 - 
425319473946139603274605151187659*x + 737707086760731113357714241006081263
> WeberClassPolynomial(-71);  
x^7 - x^6 - x^5 + x^4 - x^3 - x^2 + 2*x + 1


As in this example, a witnessed by the constant term 1, the roots of the 
WeberClassPolynomial are units in a particular ring class order. 


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