Each line is listed as:
N [A, B] [n_{1}, n_{2}] n_{1}*n_{2} j Cwhere:
We have also considered how we might divide up curves with the same conductor into isogeny classes. If two curves are isogenous, then they will have the same number of points in the residue field gotten by taking the quotient of the ring of integers by some prime ideal, so one way to divide up the curves into probable isogeny classes is just to count its points over the residue fields of a bunch of primes.
In the tables, we have chosen to record the traces of Frobenius instead of the number of point; if two curves are isogenous the trace of this map will be the same in each residue field, and the number of points in the finite field can be determined by knowing this trace.
The tables we generated follow. Each line is of the form
N*letter* [A, B] C trace of Frob_{p1} trace of Frob_{p2} ...where N is the norm of the conductor, A and B are the coefficients of the curve, and C is the actual conductor as above; and the remaining columns are the traces of the Frobenius map over the finite field corresponding to a certain prime ideal; if the elliptic curve has bad reduction over that prime ideal, we write "bad" and the trace of Frobenius is not computed. The *letter*s indicate how the curves of a given conductor norm are divided up: if the trace of the Frobenius map matches at each prime for two curves with the same conductor norm, then those curves will be listed with the same letter; for example, in the table for x^{2}+7, we have
1024A [-1,0] (32) bad bad -6 -6 0 0...
1024A [4,0] (32) bad bad -6 -6 0 0...
These match at each prime ideal we tested, so are probably in the same isogeny class. If two curves differ at some prime in the list, they are listed with different letters; for example,
16A [5,2*a] (1/2*(3*a+1)) bad bad -6 10 0 -4...
16B [5,-2*a] (1/2*(-3*a+1)) bad bad -6 10 0 4...
which comes from the same table.
The primes are generated by taking the primes in Z up through 100 and factoring them in the number field; they are not listed in the tables because we are interested in whether two curves match up at some collection of primes, so we're not actually worried which ones. The prime ideals are ordered by prime lying under them in Z; i.e., first we list primes over 2, then primes over 3, then primes over 5, etc.; however, the prime ideals over a single prime are just in the order Magma happened to print them. Thus, if you want to recreate the list of primes you could compute the trace of Frobenius over each finite field for one or two of the listed curves and match up the columns appropriately.