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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%

Distribution of Ш\(_{an} \)

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