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 eight 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 $719057$, for $n=7396$.
Credit and acknowledgements
The data files were created by Randall L. Rathbun on February 14, 2013 with updates on May 18, 2014. Some data was provided by Allan J MacLeod, Ralph Buchholz and Brett Wittby, the rest computed by Rathbun using PARI/GP and eclib (in particular mwrank).