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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 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:

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 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)
894097111327.01853534395558512966201243653950
856417111192.73058837910329021633303899432936

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).