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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 P1, while
the Weber variants define a subscheme of a modular curve of higher
level.
Given a negative discriminant D, returns the Hilbert class polynomial,
defined as the minimal polynomial of j(τ), where Z[τ] is an
imaginary quadratic order of discriminant D.
WeberClassPolynomial(D) : RngIntElt -> RngUPolElt, FldFunRatUElt
Given a negative discriminant D not congruent to 5 modulo 8,
returns the Weber class polynomial, defined as the minimal polynomial
of f(τ), where Z[τ] is an imaginary quadratic order of
discriminant D and f is a particular normalized Weber function
generating the same class field as j(τ). The particular f(τ)
used depends on whether D is odd ( ≡ 1 mod 8) or even,
in which case it also depends on D/4 mod 8, as well as whether 3
divides D or not.
A root f(τ) of
the Weber class polynomial is an algebraic integer (a unit in most cases
and always a unit outside of 2) generating the ring
class field related to the corresponding root j(τ) of the
Hilbert class polynomial by an expression j(τ) = F(f(τ)). Here
F is a rational function of the form A(Bxr + C)3/xr with r|24 and
A,B,C rational integers which are positive or negative powers of
2. The function F is also returned. For example, if
D ≡ 1 mod 8 then
j(τ) = ((f(τ)24 - 16)3 /f(τ)24),
where (GCD)(D, 3) = 1, and
j(τ) = ((f(τ)8 - 16)3 /f(τ)8),
if 3 divides D and f(τ) is a unit.
In fact, f(τ) can only be a non-unit when 4|D and
D/4 ≡ 0, 4 or 5 mod 8.
For further details, consult Yui and
Zagier [YZ97] for the case of odd D and
Schertz [Sch76] for the
case of even D.
Al: MonStgElt Default: "Roots"
Given a negative discriminant D, and the corresponding
Weber class polynomial f, returns the Hilbert class polynomial for
D. The default algorithm, as specified by the Al parameter, is
to compute complex approximations to the roots of the latter polynomial
from approximations to the roots of the Weber polynomial and the
rational function F linking the two. The other method is algebraic and
uses resultants and the function F which was discussed in the description
of the intrinsic WeberClassPolynomial.
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 indicated by the constant term -1, the roots of the
WeberClassPolynomial are units in a particular ring class order.
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