Tables of Elliptic Curves over Number Fields

Jennifer Sinnott

(A SAGE Project Advised by William Stein)

This is a table of elliptic curves over various number fields. Here is a description of how we made the tables, including the code so that the calculations can be repeated, and some discussion about how these tables compare to other people's tables.

The tables list curves over a number field K. This field is an extension of the rational numbers Q found by adjoining to Q a root of the polynomial listed as f . We let a be a root of this polynomial, so that K=Q(a). Quantities in the tables will be given as linear combinations of the power basis (1, a, ..., a^n-1) of K.

Each line is listed as:

N   [A, B]   [n₁, n₂]   n₁*n₂   j   C

where:

N is the norm of the conductor;
A and B are the coefficients in y² = x³ + Ax + B;
The torsion subgroup is given by Z/n₁Z + Z/n₂Z, and has order n₁n₂;
j is the j-invariant of the curve;
C is the actual conductor, given as a principal ideal in the ring of integers of K.

Now, here are the tables:

f = x² + 1, bound = 8: table_degree_2_poly_1_0_1_size_8.txt (41776 curves; 3.3 M)
f = x² + 2, bound = 6: table_degree_2_poly_2_0_1_size_6.txt (28548 curves; 2.1 M)
f = x² - 2, bound = 5: table_degree_2_poly_-2_0_1_size_5.txt (14626 curves; 1.1 M)
f = x² + 3, bound = 7: table_degree_2_poly_3_0_1_size_7.txt (38022 curves; 3.0 M)
f = x² - 3, bound = 5: table_degree_2_poly_-3_0_1_size_5.txt (14634 curves; 1.2 M)
f = x² - 5, bound = 7: table_degree_2_poly_-5_0_1_size_7.txt (50428 curves; 4.1 M)
f = x² - 6, bound = 6: table_degree_2_poly_-6_0_1_size_6.txt (28558 curves; 2.5 M)
f = x² + 7, bound = 5: table_degree_2_poly_7_0_1_size_5.txt (14634 curves; 1.2 M)
f = x² - 7, bound = 7: table_degree_2_poly_-7_0_1_size_7.txt (50622 curves; 4.4 M)
f = x² + 11, bound = 6: table_degree_2_poly_11_0_1_size_6.txt (28558 curves; 2.3 M)
f = x² - 11, bound = 4: table_degree_2_poly_-11_0_1_size_4.txt (6558 curves; 700 K)
f = x² - 13, bound = 4: table_degree_2_poly_-13_0_1_size_4.txt (6558 curves; 511 K)
f = x² - 14, bound = 4: table_degree_2_poly_-14_0_1_size_4.txt (6558 curves; 745 K)
f = x³ - 2, bound = 2: table_degree_3_poly_-2_0_0_1_size_2.txt (15616 curves; 1.1 M)
f = x³ + 5x + 1, bound = 2: table_degree_3_poly_1_5_0_1_size_2.txt (15624 curves; 1.9 M)
f = x⁴ + 1, bound = 1: table_degree_4_poly_1_0_0_0_1_size_1.txt (1660 curves; 249 K)
f = x⁴ + x³ + x² + x + 1, bound = 1: table_degree_4_poly_1_1_1_1_1_size_1.txt (4270 curves; 646 K)

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 x²+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.

And here are those tables; note that each one only includes curves with conductor-norm less than 100000:

f = x² + 1, bound = 5: isotable_degree_2_poly_1_0_1_size_5.txt
f = x² - 11, bound = 5: isotable_degree_2_poly_-11_0_1_size_5.txt
f = x² - 13, bound = 5: isotable_degree_2_poly_-13_0_1_size_5.txt
f = x² - 14, bound = 5: isotable_degree_2_poly_-14_0_1_size_5.txt
f = x² + 19, bound = 5: isotable_degree_2_poly_19_0_1_size_5.txt
f = x² - 2, bound = 5: isotable_degree_2_poly_-2_0_1_size_5.txt
f = x² + 2, bound = 5: isotable_degree_2_poly_2_0_1_size_5.txt
f = x² + 3, bound = 5: isotable_degree_2_poly_3_0_1_size_5.txt
f = x² + 7, bound = 5: isotable_degree_2_poly_7_0_1_size_5.txt