The database currently contains the Cremona database of all
3,064,705 elliptic curves in 2,164,260 isogeny classes,
with
conductor at most
500,000, all of which have
rank ≤ 4.
Distribution of rank
| Rank |
0 |
1 |
2 |
3 |
4 |
| number of curves |
1170876 |
1535669 |
348672 |
9487 |
1 |
| number of isogeny classes |
790258 |
1086046 |
279056 |
8899 |
1 |
| proportion |
36.51% |
50.18% |
12.89% |
0.41% |
0.00% |
Distribution of torsion
| Torsion structure |
\(C_{1}\) |
\(C_{2}\) |
\(C_{3}\) |
\(C_{4}\) |
\(C_{5}\) |
\(C_{6}\) |
\(C_{7}\) |
\(C_{8}\) |
\(C_{9}\) |
\(C_{10}\) |
\(C_{12}\) |
| Torsion order |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
12 |
| number of curves |
1683021 |
1186350 |
51405 |
28645 |
1503 |
6759 |
80 |
162 |
20 |
42 |
16 |
| proportion |
54.92% |
38.71% |
1.68% |
0.93% |
0.05% |
0.22% |
0.00% |
0.01% |
0.00% |
0.00% |
0.00% |
| |
| Torsion structure |
|
\(C_{2}\times C_{2}\) |
|
\(C_{2}\times C_{4}\) |
|
\(C_{2}\times C_{6}\) |
\(C_{2}\times C_{8}\) |
| Torsion order |
|
4 |
|
8 |
|
12 |
16 |
| number of curves |
|
104846 |
|
1753 |
|
97 |
6 |
| proportion |
|
3.42% |
|
0.06% |
|
0.00% |
0.00% |
| Ш\(_{an} \) |
\(1^2\) |
\(2^2\) |
\(3^2\) |
\(4^2\) |
\(5^2\) |
\(6^2\) |
\(7^2\) |
\(8^2\) |
\(9^2\) |
\(10^2\) |
\(11^2\) |
\(12^2\) |
| number of curves |
2821178 |
158758 |
50428 |
18170 |
8501 |
2402 |
2153 |
1301 |
595 |
315 |
300 |
145 |
| |
| Ш\(_{an} \) |
\(13^2\) |
\(14^2\) |
\(15^2\) |
\(16^2\) |
\(17^2\) |
\(18^2\) |
\(19^2\) |
\(20^2\) |
\(21^2\) |
\(22^2\) |
\(23^2\) |
\(24^2\) |
| number of curves |
160 |
58 |
43 |
45 |
34 |
7 |
29 |
15 |
5 |
2 |
13 |
4 |
| |
|
| Ш\(_{an} \) |
\(25^2\) |
\(26^2\) |
\(27^2\) |
\(28^2\) |
\(29^2\) |
\(31^2\) |
\(32^2\) |
\(33^2\) |
\(34^2\) |
\(37^2\) |
\(41^2\) |
\(43^2\) |
\(47^2\) |
\(50^2\) |
\(75^2\) |
| number of curves |
12 |
3 |
4 |
7 |
3 |
3 |
4 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
