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Author: William A. Stein
Notations and Conventions

Notations and Conventions

Armand Brumer and Oisin McGuinness

Minimal Models and Families

The notations we use for elliptic curves are conventional. We are interested in curves defined over Q, which therefore have a minimal model, defined over Z, of the form:

y2 + a1xy + a3y = x3 + a2x2 + a4x + a6

where the coefficients are all in Z, and the discriminant is minimal.

We will always assume that appropriate translations have been made so that a1 is 0 or 1; a2 is -1, 0 or 1; and a3 is 0 or 1. This is sometimes called the standard model.

We searched for our curves according to the resulting 9 families: 001, 011, 021, 100, 101, 110, 111, 120, 121; --note that the 3 other combinations (000, 010, 020) do not give minimal models of curves with prime conductor--, where "021" means, for example, a1=0, a2=-1, a3=1. (It was more convenient to use "2" instead of "-1" as part of file names.) The page "Index of Curve Files" lists the curve files organised by these families.

Curve Invariants

We were interested in computing the analytic rank, the geometric rank, and the quantities entering into the conjecture of Birch and Swinnerton-Dyer for each curve. Thus we also computed and kept the parity of the functional equation, the fundamental period, the appropriate L-value (i.e., the appropriate derivative value, depending on the known analytic rank), and the height regulator of the known points on the curve.

Without at this point going into more details on these, see "The Behavior of the Mordell-Weil Group of Elliptic Curves" for quick recall of the notations, we just say here that the usual definitions apply with some small exceptions, which we now detail.

The parity of a modular curve is either "even" (sign +1 in the functional equation of the L-series), or "odd" (sign -1 in the functional equation of the L-series). However, there are certain elliptic curves of prime conductor, the Setzer-Neumann curves, which are peculiar: they are the unique such curves with a torsion point of order 2, their conductor p is either 17 or a prime of the form u2+64, for some integer u (note it is still unknown, though expected, that there are infinitely many such primes; for such primes there exists a Setzer-Neumann curve which can be explicitly given), and they are known to be of rank 0. We decided to signal these by keeping the string "setzer" as a possible value for parity, as well as "even" and "odd". Correspondingly, all our programs that process our curve files are designed to handle this case appropriately.

Curve File Format

The curve files (see Curves), are such that each line contains curve information for a single curve, in the format illustrated by the example (from 001_3, note there is also a rank 2 curve of the same conductor in the same file):

[[0, 0, 1, -79, 342], -19047851, odd, 5, 2.047641, 30.285711, 14.790539, 5, 14.790528, [ 5, 4, 3, 7, 0] ],

From left to right we have a list of:

  • the coefficients of the minimal model (in a list),
  • the discriminant of the curve,
  • the parity,
  • the analytic rank (r),
  • the fundamental period,
  • the r-th derivative of the L-series,
  • the quotient of the last two (the "Birch-Swinnerton-Dyer number"),
  • the arithmetic rank,
  • the height regulator (of the points determined by the next item),
  • a list of the x-coordinates of a set of independent points.
(See Examples for more about this particular curve, and other examples.)

We used Maple extensively in the course of our computations, at least in the generation of curves, and prototyping phase; this format was designed to make it easy to turn a curve file into a Maple list of curves, and it was easy to parse this format in C or C++, which were used to develop the bulk of the programs that calculated.

Page created on a Macintosh Performa 6205CD, using MPW, MacPerl, and OzTeX, Sun Apr 18 23:51:20 1999.