Data for the first 1,000,000 congruent number curves is available, provided by Randall L. Rathbun. For each of the first 1,000,000 positive integers $n$, this includes information about the $n$th congruent number curve $E_n$, from which it can be determined whether or not $n$ is a congruent number, and a solution to the congruent number problem when it is.
Congruence property of specific integers
Enter an integer between 1 and 1,000,000 to see if it is a congruent number or not:
Data files
Each file consists of 1,000,000 lines, indexed by the positive integer $n$ and tab delimited.
| Download | File contents | Sample line | Notes |
|---|---|---|---|
| Conductor |
000137 60608 |
For squarefree $n$, the conductor of $E_n$ is $32n^2$ or $16n^2$, for $n$ odd and even respectively. | |
| Rank |
000137 2 |
||
| Generators |
000137 [[-3136/25,77112/125],[-38025/289,2151240/4913]] |
For three rank 2 curves, the second generator is not known and given as [?,?] | |
| Heights |
000137 [8.4621365965410101565417149184287484305,10.575571190572966907642306531966335545] |
Given to 38 decimal digit precision | |
| Regulator |
000137 84.158316595868123199826622496651245358 |
Given to 38 decimal digit precision | |
| Root number |
000137 +1 |
||
| Analytic order of Ш |
000137 1 |
Rounded value | |
| Class counts |
000137 137 1 8 |
($n$, $n_0$, $n_0\mod 8$, Heegner class number) |
Additional statistics
Distribution of rank
The distribution of ranks over the first 1,000,000 congruent number curves is as follows:
| rank | count |
|---|---|
| $0$ | 446,823 |
| $1$ | 497,599 |
| $2$ | 53,153 |
| $3$ | 2,394 |
| $4$ | 31 |
| $1\dots4$ | 553,177 |
The rank is even (root number $+1$) in $500,007$ cases, and odd (root number $-1$) in $499,993$ cases. It is positive in $553,177$ cases, and hence approximately $55.3\%$ of the first million integers are congruent numbers.
There are no curves of rank 5 or greater in this range. It has been shown (see Number Theory List Server postings #004175 and #004177 from 2011) that the first two rank 5 congruent number curves are at $n = 48,272,239$ and $n = 51,604,646$, both outside the range of this data.
In three cases ($n = 799657, 937121$, and $998617$), all of rank $2$, only one generator is known, and hence neither the regulator nor Ш are known.
Large regulators
The largest regulators are the following, both for curves of rank 2:
| $n$ | regulator (to 38 digits) |
|---|---|
| 894097 | 111327.01853534395558512966201243653950 |
| 856417 | 111192.73058837910329021633303899432936 |
Distribution of Ш
Over half the curves ($574,290$) have trivial Ш. The largest value seen is $7396 = 86^2$, for $n=719057$.
Credit and acknowledgements
The data files were created by Randall L. Rathbun on February 14, 2013 with updates on May 18, 2014. For the curves of rank one, the generators were computed by him using an implementation in PARI/GP originally by Robatino of an algorithm in Robatino's 1996 thesis Computation of mock Heegner points on modular elliptic curves, following work of Monsky. Some data was provided by Allan J. MacLeod, and by Ralph Buchholz and Brett Wittby, using Magma; the rest computed by Rathbun using PARI/GP and eclib (in particular mwrank).