Sheela Devadas

Introduction: \( \newcommand{\O}{\mathcal{O}} \newcommand{\OK}{\O_K} \newcommand{\Z}{\mathbf{Z}} \newcommand{\C}{\mathbf{C}} \newcommand{\Q}{\mathbf{Q}} \newcommand{\s}{\sqrt{-3}} \newcommand{\lr}[1]{\langle #1\rangle} \)

This page contains my computations of isomorphism classes of complex elliptic curves (or equivalently, lattices in \(\C\)) with CM by a particular order in an imaginary quadratic number field, which are in correspondence with the class group of that order. I also compute the ring class fields of these orders.

To compute the class group of an order and the homothety/isomorphism classes of lattices with CM by that order, we used the bijection between the form class group and the ideal class group described in Theorem 7.7 of [Cox].

Since Sage has implemented composition of quadratic forms and has methods to list all primitive positive definite forms of a given discriminant, the structure of the form class group can be easily computed. Then by the bijection between the form class group and the ideal class group described in Theorem 7.7 of [Cox] (namely, a primitive positive definite quadratic form \(f(x,y) = ax^2+bxy+cy^2\) of discriminant \(D\) corresponds to the proper ideal \((a,(-b+\sqrt{D})/2)\) of the unique order \(\O_D\) with discriminant \(D\)) we can then compute the homothety classes of lattices with CM by \(\O_D\).

Sage has also implemented a database of Hilbert class polynomials . If \(H_{\O}(X)\) is the Hilbert class polynomial of an order \(\O_D\), we see by Proposition 13.2 of [Cox] that by adjoining one root of \(H_{\O}(X)\) to the fraction field of \(\O_D\), we obtain the ring class field \(L_{\O}\) of \(\O_D\). By then using Sage again to factor various polynomials over this ring class field, we can obtain a simpler description of \(L_{\O}\).

Once we have a simple description of \(L_{\O}\), we can factor \(H_{\O}(X)\) over \(L_{\O}\) to get exact values of the \(j\)-invariants of the lattices with CM by \(\O_D\). To identify which \(j\)-invariant corresponds to which lattice, we need only then approximate the \(j\)-invariants of each lattice; this can be done by truncating the \(q\)-series described in Proposition 9.12 of [Wa] or in this problem set.

If you find any errors in the table below please let me know.

Bibliography:

[Cox] David A Cox. Primes of the form x^\(2\) + ny^\(2\): Fermat, class field theory, and complex multiplication. Vol. 34. John Wiley & Sons, 2011.

[Wa] Lawrence C Washington. Elliptic curves: number theory and cryptography. Chapman and Hall/CRC, 2008.

Computations:

In the tables below, which cover some orders in the imaginary quadratic number fields \(K=\Q(i), \Q(\s)\), we list the order, the class group of that order, and the ring class field. When I have computed them, I give lattices \(\lr{a,b}\) which correspond to the elements of the class group (where \(\lr{a,b}\) corresponds to the \(\Z\) span of the complex numbers \(a,b\)) and the \(j\)-invariants of those lattices.

Throughout, \(\zeta_m=e^{2\pi i/m}\), a primitive \(m\)th root of unity. The ring of integers of \(\Q(i)\) is \(\O_{\Q(i)}=\Z[i]\).

We set \(\omega=\zeta_6\). The ring of integers of \(\Q(\s)\) is \(\O_{\Q(\s)}=\Z[\frac{1+\s}{2}]=\Z[\omega]\).