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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:

e.g. 6

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:

rankcount
$0$ 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 eight cases ($n = 378953, 462073, 548953, 764593, 799657, 822722, 937121$, 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)
894097111327.01853534395558512966201243653950
856417111192.73058837910329021633303899432936

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, the rest computed by Rathbun using PARI/GP and eclib (in particular mwrank).