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Tables of Elliptic Curves over Number Fields: Methodology

Tables of Elliptic Curves over Number Fields: How We Made The Tables

The calculation was done on MECCAH using the program Magma. Some of the data on this site was generated using the original version of the program, but anyone interested in using this code for further calculations should use the new version which has been cleaned up slightly, explained better, and to which has been added the dividing up into isogeny classes bit. The structure of the program is outlined below.

Let K be a number field generated by a root a of an irreducible polynomial m(x) of degree n.

We enumerate curves defined over the number field as follows: we take curves in the form y2 = x3 + Ax + B, where A and B are elements of K written as linear combinations of the power basis (1, a, ..., an-1): so A = A0 + A1 a + ... + A n-1 an-1 and B = B0 + B1 a + ... + B n-1 an-1. We choose some integer N as a "bound," by which we mean that the coefficients Ai and Bi in the expressions for A and B are integers in the range [ -N,-N+1,...,N-1,N]

We list out all elliptic curves y2 = x3 + Ax + B with coefficients "bounded" by an integer N as described above; if two curves are isomorphic over K, we throw out one of them. Then we sort by norm of the conductor and produce the tables available on the website. For more detail, please see the new version of the code, which includes some explanatory remarks, or contact me, Jennifer Sinnott.

It may also be useful to access the data directory; if these are saved in a directory marked "newdata" the program will access them and will not need to recompute data (if it has already been computed). The first number in the name of one of these files gives the degree of the field extension, call that n. Then the next n+1 digits give the coefficients of the polynomial that generates the extension (e.g., 2_0_1 is x2+2;); and the last digit gives the "bound," N; so curves in that file are those whose coefficients have "bound" N but not "bound" N-1. This directory is currently slightly less extensive than the tables on the prior page might indicate; this is again because much of that data was computed with a slightly older version of the program, so the new version will not read it correctly; the old data directory is here; it will work with the old version of the program if saved in a directory marked "data."

Finally, these tables will hopefully be useful to people interested in elliptic curves over number fields, but it is important to make clear the strengths and weaknesses of this program; it is able to enumerate many many curves and calculate the things one might want to know about these curves; however, the way we are choosing to enumerate the curves is not exhaustive and so we miss a lot of curves. Following is a comparison to some information John Cremona calculated which will illustrate just how many curves we do miss.

We consider page 7 of the paper Modular Forms and Elliptic Curves over Imaginary Quadratic Number Fields listed on Cremona's website. The table at the bottom of that page gives the number of curves we should be finding in a selection of number fields whose conductor has norm less than some integer; complex conjugate norms are thrown out:

  • For x2+1, Cremona found 39 curves with norm-of-conductor less than 500; we found 7. (bound = 8)
  • For x2 + 2, Cremona found 36 curves with norm-of-conductor less than 300; we found 3. (bound = 6)
  • For x2+3, Cremona found 27 curves with norm-of-conductor less than 500; we found 3. (bound = 7)
  • For x2+7, Cremona found 17 curves with norm-of-conductor less than 300; we found 1. (bound = 5)
  • For x2+11, Cremona found 17 curves with norm-of-conductor less than 200; we found 0. (bound = 6)

Another table of elliptic curves over number fields can be found in Dominic Lemelin's thesis, on page 97 of the file (page 89 of the thesis). Here Lemelin gives curves with norm of conductor 297 over the field generated by a root of x2 + 2; we have no such curves. For the field generated by a root of x2 + 1, he gives a curves with norms of conductor 72 and 233 which we also do not have; and curves over x2 + 3 which we also do not have.

So, if you use these tables, it is important to keep in mind that we miss a lot of curves. On the other hand, our tables contain hundreds of thousands of curves.