explicit example of computing ray class field for imaginary quadratic?

Question

Given an imaginary quadratic number field K, we can get its ray class field mod some ideal $\mathcal{m}$ by adjoining the j-invariant of an elliptic curve with complex multiplication given by $\mathcal{O}_K$ and the values of the Weber function at the non-zero $\mathcal{m}$-torsion of the curve. (See http://www.math.leidenuniv.nl/~psh/ANTproc/15cohenpsh.pdf)

I'm trying to get my hands on understanding this construction and explicit examples would be helpful- Does anyone know of a source that really explicitly goes through this computation of the ray class field for particular examples of K and choices of $\mathcal{m}$?

Answer 1

You can do this directly for concrete examples using Sage or Magma, using Theorem 5.6 in Silverman's "Advanced topics...".

For instance, take $K=\mathbb{Q}(\sqrt{-7})$ and let us build the ray class field of conductor $3$. First, we need an elliptic curve $E$ with CM by $K$ (by $\mathcal{O}_K$). Here is one such curve: $$E/\mathbb{Q}: y^2+xy=x^3-x^2-2x-1.$$ Since $j(E)\neq 0, 1728$, the $x$-coordinate function is a Weber function (see Cor. 5.7), and since $j(E)\in\mathbb{Q}$, we can build the ray class field of conductor $3$ by adjoining to $K$ the $x$-coordinates of all $3$ torsion points. For this, we can find the third division polynomial: $$\psi_3:= 3x^4 - 3x^3 - 12x^2 - 12x - 1.$$ Let $L$ be the splitting field of $\psi_3$. Then, the ray class field of $K$ modulo $3$ is the field $H=KL$. (Note: the field $H$ turns out to be the extension $K(\alpha)$, where $\alpha$ is any root of $\psi_3$.)

You can do all this explicitly using Magma. The code is below:

E:=EllipticCurve([1,-1,0,-2,-1]);
HasComplexMultiplication(E);
P<x>:=PolynomialRing(Rationals());
f:=P!DivisionPolynomial(E,3);
f;
K:=NumberField(x^2+7);
F:=ext<K|f>;
H:=AbsoluteField(F);
H;

Above I computed $K(\alpha)$ where $\alpha$ is one $x$-coordinate of $3$ torsion. Let us calculate all the $x$-coordinates instead:

L:=SplittingField(f);
H2:=Compositum(K,L);
H2;
IsIsomorphic(H,H2);

Magma has ray class field functionality built in:

O:=MaximalOrder(K);
R:=RayClassField(3*O);
F3:=NumberField(R);
H3:=AbsoluteField(F3);
H3;
IsIsomorphic(H,H3);

Answer 2

Anyway, at http://www4.ncsu.edu/~kaltofen/bibliography/kaltofen.html :

60 E. Kaltofen and N. Yui. Explicit construction of Hilbert class fields of imaginary quadratic fields by integer lattice reduction. In D. V. Chudnovsky, G. V. Chudnovsky, H. Cohn, and M. B. Nathanson, editors, Number Theory New York Seminar 1989-1990, pages 150-202. Springer Verlag, Heidelberg, Germany, 1991. URL: EKbib/91/KaYui91.pdf.

48 E. Kaltofen, T. Valente, and N. Yui. An improved Las Vegas primality test. In Proc. 1989 Internat. Symp. Symbolic Algebraic Comput. (ISSAC'89) [-36], pages 26-33. URL: EKbib/89/KVY89.pdf.

14 E. Kaltofen and N. Yui. Explicit construction of the Hilbert class field of imaginary quadratic fields with class number 7 and 11. In Proc. EUROSAM '84 [-40], pages 310-320. URL: EKbib/84/KaYui84_eurosam.ps.gz.

12 E. Kaltofen and N. Yui. The modular equation of order 11. In Third Macsyma Users' Conference, pages 472-485. General Electric, 1984.

Note that, for my own purposes, I prefer the tables in Henri Cohen's book, Advanced Topics in Computational Number Theory. However, I am kind of a tourist, what I want is the final result, "a prime is represented by this binary quadratic form if and only if this polynomial factors into linear factors mod p."