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|
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<title>Elliptic Curve Data</title>
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<center>
<h2>
Elliptic Curve Data<br>
by J. E. Cremona<br>
University of Warwick, U.K.</h2>
<p>
<i>
Updated <a href="release_notes.md">2020-11-27</a> (last major update 2019-07-22)
</i>
</p>
</center>
<hr width="100%">
<p>
This site is a front-end to the <tt>ecdata</tt> repository, hosted at
<a href="https://github.com/JohnCremona/ecdata">GitHub</a>, which
contains data files for (modular) elliptic curves over <b>Q</b>, in a
standard format to make them easily readable by other programs. For a
typeset version of the same data (with some extra data about local
reduction data) for conductors up to 1000, you can refer to the
book <a href="http://www.warwick.ac.uk/staff/J.E.Cremona/book/amec.html">Algorithms
for modular elliptic curves </a>, CUP 1992, second revised edition
1997. See the book's web site for more information, including errata
for the current (2nd) edition, and errata to the first edition (not
maintained since the appearance of the second edition). The errata
lists include errors and omissions in the tables. The files here have
the corrected data in them. As of 2000 the book is out of print, and
CUP have no plans to reprint it.
</p>
<p>
For a more sophisticated web interface to this data and much more, use
the <a href="http://www.lmfdb.org">LMFDB</a>.
</p>
<p>
The files correspond to tables 1-5 in the book (Table 5 is not in the
First Edition), with additional tables:
<ul>
<li>Table 6 gives the isogeny matrices between curves in each isogeny
class;
<li>Table 7 lists the integral points on each curve;
<li>Table 8 gives information on which curve is optimal, and the Manin
constant;
<li>Table 9 gives information on the mod-<i>p</i> Galois
representations attached to each curve;
<li>Table 10 gives information on the 2-adic Galois
representations attached to each curve.
<li>Table 11 gives information on the growth of torsion in number
fields of small degree, for each curve.
</ul>
</p>
<p>
From September 2005, a new labelling scheme was introduced for isogeny
classes. The old scheme started
A,B,...,Z,AA,BB,...,ZZ,AAA,BBB,... and had become unwieldy. The new
scheme is a straight base 26 encoding with a=0, b=1 etc., with the
classes numbered from 0 and leading a's deleted:
a,b,...,z,ba,bb,...bz,ca,cb,... . The change to lower case is to make
codes such as bb unambiguous between the old and new systems. For
conductors less than 1728 the number of isogeny classes is at most 25
and the only change is from upper to lower case.
</p>
<p>
We give all curves in each isogeny class. For all classes of curves
of conductor less than 400000, and many others, the first one listed
in each class is proved to be the so-called "optimal" or "strong Weil"
curve attached to each newform (referred to as optimal curves from now
on). See the section <a href="#optimality">"Optimality and the Manin
constant"</a> below. Some of the data is common to all curves in the
isogeny class.
</p>
<p>
The tables currently contain data for conductors up to <b>500000</b>.
</p>
<h3>Acknowledgements</h3>
<ul>
<li>The curves were computed via modular symbols, using the C++
library eclib (see https://github.com/JohnCremona/eclib), run on
machines in a variety of places, including clusters at Nottingham
(1999-2007, conductors up to 130000); Warwick (2011-2016, conductors
130000 to 400000); and the Google Computing Platform (15-22 July 2019,
conductors 400000-500000). Thanks to the universities of Nttingham
and Warwick, and to the Simons Foundation for the GCP runs.
Additional data is computed using a SageMath script to interface a
variety of software, including the following:
</li>
<li>
The modular degrees for conductors over 12000 were computed using Mark
Watkins's programs <tt>ec</tt> and <tt>sympow</tt>, via
<tt>SageMath</tt>.
</li>
<li>
Generators for many rank 1 curves were computed using either Magma's
<tt>HeegnerPoint</tt> function (written by Mark Watkins) or
GP's <tt>ellheegner</tt> function written by Christophe Delaunay and
Bill Allombert, based on the same ideas of Delaunay and Watkins.
</li>
<li>
The integral points for all curves were first computed using Sage in
an implementation due to Michael Mardaus, Tobias Nagel and JEC. For
all curves we checked on 2018-12-19 whether these agree with Magma
(version V2.24-1): in 207 cases Magma found more integral points
(never fewer), and the lists here are now hoped to be complete.
</li>
<li>
The images of the mod <i>p</i> Galois representations were computed by
Andrew Sutherland.
</li>
<li>
The images of the 2-adic Galois representations were computed using a
Magma program provided by Jeremy Rouse.
</li>
<li>
The torsion growth data was computed by Enrique Gonzalez-Jimenez and
Filip Najman.
</li>
</ul>
<hr>
<h3>
SUMMARY TABLES
</h3>
<ul>
<li>
<form>
<select name="url" width=30>
<option value="None">Select a conductor range</option>
count/count.00000-09999
<option value="count/count.00000-09999">1-9999</option>
<option value="count/count.10000-19999">10000-19999</option>
<option value="count/count.20000-29999">20000-29999</option>
<option value="count/count.30000-39999">30000-39999</option>
<option value="count/count.40000-49999">40000-49999</option>
<option value="count/count.50000-59999">50000-59999</option>
<option value="count/count.60000-69999">60000-69999</option>
<option value="count/count.70000-79999">70000-79999</option>
<option value="count/count.80000-89999">80000-89999</option>
<option value="count/count.90000-99999">90000-99999</option>
<option value="count/count.100000-109999">100000-109999</option>
<option value="count/count.110000-119999">110000-119999</option>
<option value="count/count.120000-129999">120000-129999</option>
<option value="count/count.130000-139999">130000-139999</option>
<option value="count/count.140000-149999">140000-149999</option>
<option value="count/count.150000-159999">150000-159999</option>
<option value="count/count.160000-169999">160000-169999</option>
<option value="count/count.170000-179999">170000-179999</option>
<option value="count/count.180000-189999">180000-189999</option>
<option value="count/count.190000-199999">190000-199999</option>
<option value="count/count.200000-209999">200000-209999</option>
<option value="count/count.210000-219999">210000-219999</option>
<option value="count/count.220000-229999">220000-229999</option>
<option value="count/count.230000-239999">230000-239999</option>
<option value="count/count.240000-249999">240000-249999</option>
<option value="count/count.250000-259999">250000-259999</option>
<option value="count/count.260000-269999">260000-269999</option>
<option value="count/count.270000-279999">270000-279999</option>
<option value="count/count.280000-289999">280000-289999</option>
<option value="count/count.290000-299999">290000-299999</option>
<option value="count/count.300000-309999">300000-309999</option>
<option value="count/count.310000-319999">310000-319999</option>
<option value="count/count.320000-329999">320000-329999</option>
<option value="count/count.330000-339999">330000-339999</option>
<option value="count/count.340000-349999">340000-349999</option>
<option value="count/count.350000-359999">350000-359999</option>
<option value="count/count.360000-369999">360000-369999</option>
<option value="count/count.370000-379999">370000-379999</option>
<option value="count/count.380000-389999">380000-389999</option>
<option value="count/count.390000-399999">390000-399999</option>
<option value="count/count.400000-409999">400000-409999</option>
<option value="count/count.410000-419999">410000-419999</option>
<option value="count/count.420000-429999">420000-429999</option>
<option value="count/count.430000-439999">430000-439999</option>
<option value="count/count.440000-449999">440000-449999</option>
<option value="count/count.450000-459999">450000-459999</option>
<option value="count/count.460000-469999">460000-469999</option>
<option value="count/count.470000-479999">470000-479999</option>
<option value="count/count.480000-489999">480000-489999</option>
<option value="count/count.490000-499999">490000-499999</option>
</select>
<input type=button value="Fetch" onClick="JumpToIt(this.form)">
</form>
Lists of number of curves (up to isogeny) of each individual conductor
(each file is 10000 lines long).
</li>
<li> <a href="table.html">Table</a> showing number of curves of each
range of 10000 conductors, sorted by rank.
</li>
</ul>
<h3>
TABLE ONE: CURVES
</h3>
<ul>
<li>
<form>
<select name="url" width=30>
<option value="None">Select a conductor range</option>
<option value="allcurves/allcurves.00000-09999">1-9999</option>
<option value="allcurves/allcurves.10000-19999">10000-19999</option>
<option value="allcurves/allcurves.20000-29999">20000-29999</option>
<option value="allcurves/allcurves.30000-39999">30000-39999</option>
<option value="allcurves/allcurves.40000-49999">40000-49999</option>
<option value="allcurves/allcurves.50000-59999">50000-59999</option>
<option value="allcurves/allcurves.60000-69999">60000-69999</option>
<option value="allcurves/allcurves.70000-79999">70000-79999</option>
<option value="allcurves/allcurves.80000-89999">80000-89999</option>
<option value="allcurves/allcurves.90000-99999">90000-99999</option>
<option value="allcurves/allcurves.100000-109999">100000-109999</option>
<option value="allcurves/allcurves.110000-119999">110000-119999</option>
<option value="allcurves/allcurves.120000-129999">120000-129999</option>
<option value="allcurves/allcurves.130000-139999">130000-139999</option>
<option value="allcurves/allcurves.140000-149999">140000-149999</option>
<option value="allcurves/allcurves.150000-159999">150000-159999</option>
<option value="allcurves/allcurves.160000-169999">160000-169999</option>
<option value="allcurves/allcurves.170000-179999">170000-179999</option>
<option value="allcurves/allcurves.180000-189999">180000-189999</option>
<option value="allcurves/allcurves.190000-199999">190000-199999</option>
<option value="allcurves/allcurves.200000-209999">200000-209999</option>
<option value="allcurves/allcurves.210000-219999">210000-219999</option>
<option value="allcurves/allcurves.220000-229999">220000-229999</option>
<option value="allcurves/allcurves.230000-239999">230000-239999</option>
<option value="allcurves/allcurves.240000-249999">240000-249999</option>
<option value="allcurves/allcurves.250000-259999">250000-259999</option>
<option value="allcurves/allcurves.260000-269999">260000-269999</option>
<option value="allcurves/allcurves.270000-279999">270000-279999</option>
<option value="allcurves/allcurves.280000-289999">280000-289999</option>
<option value="allcurves/allcurves.290000-299999">290000-299999</option>
<option value="allcurves/allcurves.300000-309999">300000-309999</option>
<option value="allcurves/allcurves.310000-319999">310000-319999</option>
<option value="allcurves/allcurves.320000-329999">320000-329999</option>
<option value="allcurves/allcurves.330000-339999">330000-339999</option>
<option value="allcurves/allcurves.340000-349999">340000-349999</option>
<option value="allcurves/allcurves.350000-359999">350000-359999</option>
<option value="allcurves/allcurves.360000-369999">360000-369999</option>
<option value="allcurves/allcurves.370000-379999">370000-379999</option>
<option value="allcurves/allcurves.380000-389999">380000-389999</option>
<option value="allcurves/allcurves.390000-399999">390000-399999</option>
<option value="allcurves/allcurves.400000-409999">400000-409999</option>
<option value="allcurves/allcurves.410000-419999">410000-419999</option>
<option value="allcurves/allcurves.420000-429999">420000-429999</option>
<option value="allcurves/allcurves.430000-439999">430000-439999</option>
<option value="allcurves/allcurves.440000-449999">440000-449999</option>
<option value="allcurves/allcurves.450000-459999">450000-459999</option>
<option value="allcurves/allcurves.460000-469999">460000-469999</option>
<option value="allcurves/allcurves.470000-479999">470000-479999</option>
<option value="allcurves/allcurves.480000-489999">480000-489999</option>
<option value="allcurves/allcurves.490000-499999">490000-499999</option>
</select>
<input type=button value="Fetch" onClick="JumpToIt(this.form)">
</form>
<p>
One entry for each isomorphism class of curves, giving conductor N,
letter id for isogeny class, number of the curve in the class,
coefficients of minimal Weierstrass equation, rank r, order of torsion
subgroup |T|. For all N up to 370000 the optimal
Γ<sub>0</sub>(N) curve is the one labelled 1 (except for class
990h when it is the curve labelled 3). For N>370000, this is
probably also true, but in some cases remains conditional on Stevens'
Conjecture (see the section <a href="#optimality">"Optimality and the
Manin constant"</a> below).
</p>
<p>
Data format with sample line:
<table cellspacing="10" border="1">
<tbody><tr>
<th>N</th> <th>C</th> <th>#</th> <th>curve</th> <th>r</th> <th>t</th>
</tr>
<tr>
<td>2730</td> <td>bd</td> <td>1</td> <td>[1,0,0,-25725,1577457]</td>
<td>0</td> <td>12</td> </tr>
</tbody></table>
where:
</p><ul>
<li> <b>N</b> = conductor
</li><li> <b>C</b> = isogeny class (letter(s))
</li><li> <b>#</b> = number of curve in class = 1 (except for 990h3)
</li><li> <b>curve</b> = curve coefficients in format [a1,a2,a3,a4,a6]
</li><li> <b>r</b> = rank
</li><li> <b>t</b> = order of torsion
</li></ul>
<p>
Simple searches may be carried out with the unix/linux utility awk.
For example:
</p><p>
</p><ul>
<li>All curves with torsion of order 12:
<p align="LEFT"><tt> awk '$6==12' allcurves.* | sort -n
-k 1</tt></p>
</li><li>All curves with torsion of order 16:
<p align="LEFT"><tt> awk '$6==16' allcurves.*</tt></p>
</li><li>All curves of rank 3:
<p align="LEFT"><tt> awk '$5==3' allcurves.* | sort -n
-k 1 </tt></p>
</li></ul>
If it is desired to have the curve coefficients in five separate
fields with spaces as field separators, this can be achieved using
scripts such as this: <p align="LEFT"><tt> sed
's/[]\[,]/ /g' allcurves.00000-10000</tt></p>
<!--
<LI><A HREF="curves.1-1000.html">curves.1-1000.html</A>, experimental
format for a table in html.
-->
</li></ul>
<h3>
TABLE TWO: GENERATORS
</h3>
<ul>
<li>
<form>
<select name="url" width=30>
<option value="None">Select a conductor range</option>
<option value="allgens/allgens.00000-09999">1-9999</option>
<option value="allgens/allgens.10000-19999">10000-19999</option>
<option value="allgens/allgens.20000-29999">20000-29999</option>
<option value="allgens/allgens.30000-39999">30000-39999</option>
<option value="allgens/allgens.40000-49999">40000-49999</option>
<option value="allgens/allgens.50000-59999">50000-59999</option>
<option value="allgens/allgens.60000-69999">60000-69999</option>
<option value="allgens/allgens.70000-79999">70000-79999</option>
<option value="allgens/allgens.80000-89999">80000-89999</option>
<option value="allgens/allgens.90000-99999">90000-99999</option>
<option value="allgens/allgens.100000-109999">100000-109999</option>
<option value="allgens/allgens.110000-119999">110000-119999</option>
<option value="allgens/allgens.120000-129999">120000-129999</option>
<option value="allgens/allgens.130000-139999">130000-139999</option>
<option value="allgens/allgens.140000-149999">140000-149999</option>
<option value="allgens/allgens.150000-159999">150000-159999</option>
<option value="allgens/allgens.160000-169999">160000-169999</option>
<option value="allgens/allgens.170000-179999">170000-179999</option>
<option value="allgens/allgens.180000-189999">180000-189999</option>
<option value="allgens/allgens.190000-199999">190000-199999</option>
<option value="allgens/allgens.200000-209999">200000-209999</option>
<option value="allgens/allgens.210000-219999">210000-219999</option>
<option value="allgens/allgens.220000-229999">220000-229999</option>
<option value="allgens/allgens.230000-239999">230000-239999</option>
<option value="allgens/allgens.240000-249999">240000-249999</option>
<option value="allgens/allgens.250000-259999">250000-259999</option>
<option value="allgens/allgens.260000-269999">260000-269999</option>
<option value="allgens/allgens.270000-279999">270000-279999</option>
<option value="allgens/allgens.280000-289999">280000-289999</option>
<option value="allgens/allgens.290000-299999">290000-299999</option>
<option value="allgens/allgens.300000-309999">300000-309999</option>
<option value="allgens/allgens.310000-319999">310000-319999</option>
<option value="allgens/allgens.320000-329999">320000-329999</option>
<option value="allgens/allgens.330000-339999">330000-339999</option>
<option value="allgens/allgens.340000-349999">340000-349999</option>
<option value="allgens/allgens.350000-359999">350000-359999</option>
<option value="allgens/allgens.360000-369999">360000-369999</option>
<option value="allgens/allgens.370000-379999">370000-379999</option>
<option value="allgens/allgens.380000-389999">380000-389999</option>
<option value="allgens/allgens.390000-399999">390000-399999</option>
<option value="allgens/allgens.400000-409999">400000-409999</option>
<option value="allgens/allgens.410000-419999">410000-419999</option>
<option value="allgens/allgens.420000-429999">420000-429999</option>
<option value="allgens/allgens.430000-439999">430000-439999</option>
<option value="allgens/allgens.440000-449999">440000-449999</option>
<option value="allgens/allgens.450000-459999">450000-459999</option>
<option value="allgens/allgens.460000-469999">460000-469999</option>
<option value="allgens/allgens.470000-479999">470000-479999</option>
<option value="allgens/allgens.480000-489999">480000-489999</option>
<option value="allgens/allgens.490000-499999">490000-499999</option>
</select>
<input type=button value="Fetch" onClick="JumpToIt(this.form)">
</form>
<p>
For every curve, generators are given for the Mordell group, in
projective coordinates. <b>N.B.</b> In
<b>all</b> cases I have checked that the point(s) given are indeed
generators. Each entry consists of conductor N, isogeny class code,
number of curve in class, curve coefficients, rank r, torsion
structure (as a list of t structure constants for t=0,1 or 2, i.e. in
the form [] or [t] or [t1,t2]) and r+t points in projective
coordinates (torsion last). For example, the entry
</p>
<table cellspacing="20" border="1">
<tbody><tr>
<td>389</td><td>a</td><td>1</td><td>[0,1,1,-2,0]</td><td>2</td><td>[]</td><td>[0:0:1]</td><td>[1:0:1]
</td></tr>
</tbody></table>
<p>
means that curve 389a1 = [0,1,1,-2,0] has rank 2 and trivial torsion, with generators [0:0:1]=(0,0)
and [1:0:1]=(1,0), while the entry
</p>
<table cellspacing="2" border="1">
<tbody><tr>
<td>4602</td>
<td>a</td>
<td>1</td>
<td>[1,1,0,-37746035,-89296920339]</td>
<td>1</td>
<td>[2]</td>
<td>[175781888357266265777015693706802984972253428834450486976370 : 19575260230015313702261379022151675961965157108920263594545223 : 11451799510178287699130942513632433218384249076487302907]</td>
<td>[7094:-3547:1]</td>
</tr>
</tbody></table>
<p>
means that curve 4602a1 = [1,1,0,-37746035,-89296920339] has rank 1 with
generator
</p><pre> 77985922458974949246858229195945103471590 19575260230015313702261379022151675961965157108920263594545223
[----------------------------------------- , -------------------------------------------------------------- ]
2254020761884782243^2 2254020761884782243^3
</pre>
together with torsion of order 2 generated by [7094:-3547:1] = (7094,-3547).
<p>
<b>N.B.</b> From April 2011 the format of these files was changed to
include information about the torsion; there is therefore now a line
in the allgens files for every curve, not just those of positive rank. The
files for N<130000 were updated accordingly on 15/4/11.
</p>
</li></ul>
<h3>
TABLE THREE: HECKE EIGENVALUES
</h3>
<ul>
<li>
<form>
<select name="url" width=30>
<option value="None">Select a conductor range</option>
<option value="aplist/aplist.00000-09999">1-9999</option>
<option value="aplist/aplist.10000-19999">10000-19999</option>
<option value="aplist/aplist.20000-29999">20000-29999</option>
<option value="aplist/aplist.30000-39999">30000-39999</option>
<option value="aplist/aplist.40000-49999">40000-49999</option>
<option value="aplist/aplist.50000-59999">50000-59999</option>
<option value="aplist/aplist.60000-69999">60000-69999</option>
<option value="aplist/aplist.70000-79999">70000-79999</option>
<option value="aplist/aplist.80000-89999">80000-89999</option>
<option value="aplist/aplist.90000-99999">90000-99999</option>
<option value="aplist/aplist.100000-109999">100000-109999</option>
<option value="aplist/aplist.110000-119999">110000-119999</option>
<option value="aplist/aplist.120000-129999">120000-129999</option>
<option value="aplist/aplist.130000-139999">130000-139999</option>
<option value="aplist/aplist.140000-149999">140000-149999</option>
<option value="aplist/aplist.150000-159999">150000-159999</option>
<option value="aplist/aplist.160000-169999">160000-169999</option>
<option value="aplist/aplist.170000-179999">170000-179999</option>
<option value="aplist/aplist.180000-189999">180000-189999</option>
<option value="aplist/aplist.190000-199999">190000-199999</option>
<option value="aplist/aplist.200000-209999">200000-209999</option>
<option value="aplist/aplist.210000-219999">210000-219999</option>
<option value="aplist/aplist.220000-229999">220000-229999</option>
<option value="aplist/aplist.230000-239999">230000-239999</option>
<option value="aplist/aplist.240000-249999">240000-249999</option>
<option value="aplist/aplist.250000-259999">250000-259999</option>
<option value="aplist/aplist.260000-269999">260000-269999</option>
<option value="aplist/aplist.270000-279999">270000-279999</option>
<option value="aplist/aplist.280000-289999">280000-289999</option>
<option value="aplist/aplist.290000-299999">290000-299999</option>
<option value="aplist/aplist.300000-309999">300000-309999</option>
<option value="aplist/aplist.310000-319999">310000-319999</option>
<option value="aplist/aplist.320000-329999">320000-329999</option>
<option value="aplist/aplist.330000-339999">330000-339999</option>
<option value="aplist/aplist.340000-349999">340000-349999</option>
<option value="aplist/aplist.350000-359999">350000-359999</option>
<option value="aplist/aplist.360000-369999">360000-369999</option>
<option value="aplist/aplist.370000-379999">370000-379999</option>
<option value="aplist/aplist.380000-389999">380000-389999</option>
<option value="aplist/aplist.390000-399999">390000-399999</option>
<option value="aplist/aplist.400000-409999">400000-409999</option>
<option value="aplist/aplist.410000-419999">410000-419999</option>
<option value="aplist/aplist.420000-429999">420000-429999</option>
<option value="aplist/aplist.430000-439999">430000-439999</option>
<option value="aplist/aplist.440000-449999">440000-449999</option>
<option value="aplist/aplist.450000-459999">450000-459999</option>
<option value="aplist/aplist.460000-469999">460000-469999</option>
<option value="aplist/aplist.470000-479999">470000-479999</option>
<option value="aplist/aplist.480000-489999">480000-489999</option>
<option value="aplist/aplist.490000-499999">490000-499999</option>
</select>
<input type=button value="Fetch" onClick="JumpToIt(this.form)">
</form>
Hecke eigenvalues for p<100 for each of the corresponding newforms for
Γ<sub>0</sub>(N). When p|N the entry is simply "+" or "-" and is a W-eigenvalue,
as in Antwerp IV. When there are primes p|n with p>100 the corresponding
eigenvalue(s) are in extra column(s), as in
<br>
<table cellspacing="10" border="1">
<tbody><tr>
<td>101</td><td>a</td><td>0</td><td>-2</td><td>-1</td><td>-2</td><td>-2</td><td>1</td><td>3</td><td>-5</td><td>1
</td><td>-4</td><td>-9</td><td>-2</td><td>8</td><td>-8</td><td>7</td><td>-2
</td><td>-14</td><td>4</td><td>2</td><td>13</td><td>8</td><td>-9</td><td>-4
</td><td>14</td><td>2</td><td>+(101)
</td></tr></tbody></table>
<br>
<table cellspacing="10" border="1">
<tbody><tr>
<td>10201</td><td>a</td><td>0</td><td>2</td><td>-1</td><td>2</td><td>2</td><td>1</td><td>3</td><td>-5</td><td>
1</td><td>4</td><td>-9</td><td>-2</td><td>-8</td><td>-8</td><td>7</td><td>2</td><td>14</td><td>-4</td><td>
-2</td><td>13</td><td>-8</td><td>-9</td><td>4</td><td>-14</td><td>2</td><td>+(101)</td></tr></tbody></table>
<br>
<table cellspacing="10" border="1">
<tbody><tr>
<td>19153</td><td>a</td><td>2</td><td>0</td><td>-1</td><td>0</td><td>-4</td><td>7</td><td>-3</td><td>-3</td><td>-6</td><td>3</td><td>8</td><td>-2</td><td>0</td><td>1</td><td>1</td><td>0</td><td>15</td><td>6</td><td>
-13</td><td>12</td><td>-2</td><td>2</td><td>9</td><td>-9</td><td>-10</td><td>+(107)</td><td>-(179)
</td></tr></tbody></table>
<br>
so the total number of fields is 27, 28 or 29 on each line (assuming
N<1113121=101*103*107)
</li></ul>
<h3>
TABLE FOUR: BSD DATA and ANALYTIC ORDERS OF SHA
</h3>
<ul>
<li>
<a href="https://raw.githubusercontent.com/JohnCremona/ecdata/master/allbsd/bsd.1-1000">1-1000</a>
<p>
Birch--Swinnerton-Dyer data for the optimal
curve in each class, exactly as in the book. Column headings:
Conductor, class id letter, rank, real period w, L^(r)(1)/r!,
regulator R, rational factor, S. Here the rational factor is
L^(r)(1)/wRr!; when r=0 this is exact and given as a pair of integers
(numerator denominator); when r>0 it is approximate, but easily
recognizable. Lastly, S is the value of the order of the
Tate-Shafarevich group as predicted by B-SD (the "analytic order of
Sha"), given the previous data and also the local factors and
torsion. When r=0 this is exact; when r>0 it is approximate, and was
computed to several places but to save space is just entered as
1.0. (S>1 in only 4 cases, where S=4 or 9).
</p>
</li><li>
<form>
<select name="url" width=30>
<option value="None">Select a conductor range</option>
<option value="allbsd/allbsd.00000-09999">1-9999</option>
<option value="allbsd/allbsd.10000-19999">10000-19999</option>
<option value="allbsd/allbsd.20000-29999">20000-29999</option>
<option value="allbsd/allbsd.30000-39999">30000-39999</option>
<option value="allbsd/allbsd.40000-49999">40000-49999</option>
<option value="allbsd/allbsd.50000-59999">50000-59999</option>
<option value="allbsd/allbsd.60000-69999">60000-69999</option>
<option value="allbsd/allbsd.70000-79999">70000-79999</option>
<option value="allbsd/allbsd.80000-89999">80000-89999</option>
<option value="allbsd/allbsd.90000-99999">90000-99999</option>
<option value="allbsd/allbsd.100000-109999">100000-109999</option>
<option value="allbsd/allbsd.110000-119999">110000-119999</option>
<option value="allbsd/allbsd.120000-129999">120000-129999</option>
<option value="allbsd/allbsd.130000-139999">130000-139999</option>
<option value="allbsd/allbsd.140000-149999">140000-149999</option>
<option value="allbsd/allbsd.150000-159999">150000-159999</option>
<option value="allbsd/allbsd.160000-169999">160000-169999</option>
<option value="allbsd/allbsd.170000-179999">170000-179999</option>
<option value="allbsd/allbsd.180000-189999">180000-189999</option>
<option value="allbsd/allbsd.190000-199999">190000-199999</option>
<option value="allbsd/allbsd.200000-209999">200000-209999</option>
<option value="allbsd/allbsd.210000-219999">210000-219999</option>
<option value="allbsd/allbsd.220000-229999">220000-229999</option>
<option value="allbsd/allbsd.230000-239999">230000-239999</option>
<option value="allbsd/allbsd.240000-249999">240000-249999</option>
<option value="allbsd/allbsd.250000-259999">250000-259999</option>
<option value="allbsd/allbsd.260000-269999">260000-269999</option>
<option value="allbsd/allbsd.270000-279999">270000-279999</option>
<option value="allbsd/allbsd.280000-289999">280000-289999</option>
<option value="allbsd/allbsd.290000-299999">290000-299999</option>
<option value="allbsd/allbsd.300000-309999">300000-309999</option>
<option value="allbsd/allbsd.310000-319999">310000-319999</option>
<option value="allbsd/allbsd.320000-329999">320000-329999</option>
<option value="allbsd/allbsd.330000-339999">330000-339999</option>
<option value="allbsd/allbsd.340000-349999">340000-349999</option>
<option value="allbsd/allbsd.350000-359999">350000-359999</option>
<option value="allbsd/allbsd.360000-369999">360000-369999</option>
<option value="allbsd/allbsd.370000-379999">370000-379999</option>
<option value="allbsd/allbsd.380000-389999">380000-389999</option>
<option value="allbsd/allbsd.390000-399999">390000-399999</option>
<option value="allbsd/allbsd.400000-409999">400000-409999</option>
<option value="allbsd/allbsd.410000-419999">410000-419999</option>
<option value="allbsd/allbsd.420000-429999">420000-429999</option>
<option value="allbsd/allbsd.430000-439999">430000-439999</option>
<option value="allbsd/allbsd.440000-449999">440000-449999</option>
<option value="allbsd/allbsd.450000-459999">450000-459999</option>
<option value="allbsd/allbsd.460000-469999">460000-469999</option>
<option value="allbsd/allbsd.470000-479999">470000-479999</option>
<option value="allbsd/allbsd.480000-489999">480000-489999</option>
<option value="allbsd/allbsd.490000-499999">490000-499999</option>
</select>
<input type=button value="Fetch" onClick="JumpToIt(this.form)">
</form>
Same as previous but with data for all the curves (not only the
optimal ones) up to the current bound. <br>
Data format with sample lines:
<table cellspacing="10" border="1">
<tbody><tr>
<th>N</th><th>C</th><th>#</th><th>curve</th><th>r</th><th>t</th>
<th>cp</th><th>om</th><th>L</th><th>R</th><th>S</th>
</tr>
<tr>
<td>11</td> <td>a</td> <td>1</td> <td>[0,-1,1,-10,-20]</td> <td>0</td> <td>5</td>
<td>5</td> <td>1.269209304</td> <td>0.25384186</td> <td>1</td> <td>1</td>
</tr>
<tr>
<td>5077</td> <td>a</td> <td>1</td> <td>[0,0,1,-7,6]</td> <td>3</td> <td>1</td>
<td>1</td> <td>4.151687983</td> <td>1.73184990</td> <td> 0.41714355</td> <td>1.00000000</td>
</tr>
</tbody></table>
where:
<ul>
<li> <b>N</b> = conductor
</li><li> <b>C</b> = isogeny class (letter(s))
</li><li> <b>#</b> = number of curve in class
</li><li> <b>curve</b> = curve coefficients in format [a1,a2,a3,a4,a6]
</li><li> <b>r</b> = rank
</li><li> <b>t</b> = order of torsion
</li><li> <b>cp</b> = product of Tamagawa factors <i>c<sub>p</sub></i>
</li><li> <b>om</b> = real period
</li><li> <b>L</b> = <i>L<sup>(r)</sup>(E,1)</i>/<i>r</i>!.
</li><li> <b>R</b> = Regulator
</li><li> <b>S</b> = (Analytic) order of Sha.
</li></ul>
</li>
<br>
<li>
<form>
<select name="url" width=30>
<option value="None">Select a conductor range</option>
<option value="allbigsha/allbigsha.00000-09999">1-9999</option>
<option value="allbigsha/allbigsha.10000-19999">10000-19999</option>
<option value="allbigsha/allbigsha.20000-29999">20000-29999</option>
<option value="allbigsha/allbigsha.30000-39999">30000-39999</option>
<option value="allbigsha/allbigsha.40000-49999">40000-49999</option>
<option value="allbigsha/allbigsha.50000-59999">50000-59999</option>
<option value="allbigsha/allbigsha.60000-69999">60000-69999</option>
<option value="allbigsha/allbigsha.70000-79999">70000-79999</option>
<option value="allbigsha/allbigsha.80000-89999">80000-89999</option>
<option value="allbigsha/allbigsha.90000-99999">90000-99999</option>
<option value="allbigsha/allbigsha.100000-109999">100000-109999</option>
<option value="allbigsha/allbigsha.110000-119999">110000-119999</option>
<option value="allbigsha/allbigsha.120000-129999">120000-129999</option>
<option value="allbigsha/allbigsha.130000-139999">130000-139999</option>
<option value="allbigsha/allbigsha.140000-149999">140000-149999</option>
<option value="allbigsha/allbigsha.150000-159999">150000-159999</option>
<option value="allbigsha/allbigsha.160000-169999">160000-169999</option>
<option value="allbigsha/allbigsha.170000-179999">170000-179999</option>
<option value="allbigsha/allbigsha.180000-189999">180000-189999</option>
<option value="allbigsha/allbigsha.190000-199999">190000-199999</option>
<option value="allbigsha/allbigsha.200000-209999">200000-209999</option>
<option value="allbigsha/allbigsha.210000-219999">210000-219999</option>
<option value="allbigsha/allbigsha.220000-229999">220000-229999</option>
<option value="allbigsha/allbigsha.230000-239999">230000-239999</option>
<option value="allbigsha/allbigsha.240000-249999">240000-249999</option>
<option value="allbigsha/allbigsha.250000-259999">250000-259999</option>
<option value="allbigsha/allbigsha.260000-269999">260000-269999</option>
<option value="allbigsha/allbigsha.270000-279999">270000-279999</option>
<option value="allbigsha/allbigsha.280000-289999">280000-289999</option>
<option value="allbigsha/allbigsha.290000-299999">290000-299999</option>
<option value="allbigsha/allbigsha.300000-309999">300000-309999</option>
<option value="allbigsha/allbigsha.310000-319999">310000-319999</option>
<option value="allbigsha/allbigsha.320000-329999">320000-329999</option>
<option value="allbigsha/allbigsha.330000-339999">330000-339999</option>
<option value="allbigsha/allbigsha.340000-349999">340000-349999</option>
<option value="allbigsha/allbigsha.350000-359999">350000-359999</option>
<option value="allbigsha/allbigsha.360000-369999">360000-369999</option>
<option value="allbigsha/allbigsha.370000-379999">370000-379999</option>
<option value="allbigsha/allbigsha.380000-389999">380000-389999</option>
<option value="allbigsha/allbigsha.390000-399999">390000-399999</option>
<option value="allbigsha/allbigsha.400000-409999">400000-409999</option>
<option value="allbigsha/allbigsha.410000-419999">410000-419999</option>
<option value="allbigsha/allbigsha.420000-429999">420000-429999</option>
<option value="allbigsha/allbigsha.430000-439999">430000-439999</option>
<option value="allbigsha/allbigsha.440000-449999">440000-449999</option>
<option value="allbigsha/allbigsha.450000-459999">450000-459999</option>
<option value="allbigsha/allbigsha.460000-469999">460000-469999</option>
<option value="allbigsha/allbigsha.470000-479999">470000-479999</option>
<option value="allbigsha/allbigsha.480000-489999">480000-489999</option>
<option value="allbigsha/allbigsha.490000-499999">490000-499999</option>
</select>
<input type=button value="Fetch" onClick="JumpToIt(this.form)">
</form>
Lists of the curves with non-trivial Tate-Shafarevich group, according
to the BSD conjecture; i.e., curves whose "analytic order of Sha" is
greater than 1. The record (to conductor 410000) is 5625=75<sup>2</sup>
for 165066d3.
</li>
<br>
<li>
<a href="shas.html">shas.html</a><br>
A summary table of large Shas. Note that up to conductor 500000
there are 243527 elliptic curves with non-trivial Sha, with ranks 0
(222922 curves), 1 (20563 curves), 2 (42 curves, all with
2-torsion).
</li></ul>
<h3>
TABLE FIVE: PARAMETRIZATION DEGREES
</h3>
<ul>
<li>
Optimal curves:
<form>
<select name="url" width=30>
<option value="None">Select a conductor range</option>
<option value="degphi/degphi.00000-09999">1-9999</option>
<option value="degphi/degphi.10000-19999">10000-19999</option>
<option value="degphi/degphi.20000-29999">20000-29999</option>
<option value="degphi/degphi.30000-39999">30000-39999</option>
<option value="degphi/degphi.40000-49999">40000-49999</option>
<option value="degphi/degphi.50000-59999">50000-59999</option>
<option value="degphi/degphi.60000-69999">60000-69999</option>
<option value="degphi/degphi.70000-79999">70000-79999</option>
<option value="degphi/degphi.80000-89999">80000-89999</option>
<option value="degphi/degphi.90000-99999">90000-99999</option>
<option value="degphi/degphi.100000-109999">100000-109999</option>
<option value="degphi/degphi.110000-119999">110000-119999</option>
<option value="degphi/degphi.120000-129999">120000-129999</option>
<option value="degphi/degphi.130000-139999">130000-139999</option>
<option value="degphi/degphi.140000-149999">140000-149999</option>
<option value="degphi/degphi.150000-159999">150000-159999</option>
<option value="degphi/degphi.160000-169999">160000-169999</option>
<option value="degphi/degphi.170000-179999">170000-179999</option>
<option value="degphi/degphi.180000-189999">180000-189999</option>
<option value="degphi/degphi.190000-199999">190000-199999</option>
<option value="degphi/degphi.200000-209999">200000-209999</option>
<option value="degphi/degphi.210000-219999">210000-219999</option>
<option value="degphi/degphi.220000-229999">220000-229999</option>
<option value="degphi/degphi.230000-239999">230000-239999</option>
<option value="degphi/degphi.240000-249999">240000-249999</option>
<option value="degphi/degphi.250000-259999">250000-259999</option>
<option value="degphi/degphi.260000-269999">260000-269999</option>
<option value="degphi/degphi.270000-279999">270000-279999</option>
<option value="degphi/degphi.280000-289999">280000-289999</option>
<option value="degphi/degphi.290000-299999">290000-299999</option>
<option value="degphi/degphi.300000-309999">300000-309999</option>
<option value="degphi/degphi.310000-319999">310000-319999</option>
<option value="degphi/degphi.320000-329999">320000-329999</option>
<option value="degphi/degphi.330000-339999">330000-339999</option>
<option value="degphi/degphi.340000-349999">340000-349999</option>
<option value="degphi/degphi.350000-359999">350000-359999</option>
<option value="degphi/degphi.360000-369999">360000-369999</option>
<option value="degphi/degphi.370000-379999">370000-379999</option>
<option value="degphi/degphi.380000-389999">380000-389999</option>
<option value="degphi/degphi.390000-399999">390000-399999</option>
<option value="degphi/degphi.400000-409999">400000-409999</option>
<option value="degphi/degphi.410000-419999">410000-419999</option>
<option value="degphi/degphi.420000-429999">420000-429999</option>
<option value="degphi/degphi.430000-439999">430000-439999</option>
<option value="degphi/degphi.440000-449999">440000-449999</option>
<option value="degphi/degphi.450000-459999">450000-459999</option>
<option value="degphi/degphi.460000-469999">460000-469999</option>
<option value="degphi/degphi.470000-479999">470000-479999</option>
<option value="degphi/degphi.480000-489999">480000-489999</option>
<option value="degphi/degphi.490000-499999">490000-499999</option>
</select>
<input type=button value="Fetch" onClick="JumpToIt(this.form)">
</form>
A table of the degree of the modular parametrizations of each optimal curve.
<br>
Data format with sample line:
<table cellspacing="10" border="1">
<tbody><tr>
<th>N</th> <th>id</th> <th>degree</th> <th>primes</th> <th>curve</th>
</tr>
<tr>
<td>5077</td> <td>a 1</td> <td>1984</td> <td>{2,31}</td> <td>[0,0,1,-7,6]</td>
</tr></tbody></table>
where "primes" is the set of primes dividing the degree.
</li>
<br>
<li>
All curves:
<form>
<select name="url" width=30>
<option value="None">Select a conductor range</option>
<option value="alldegphi/alldegphi.00000-09999">1-9999</option>
<option value="alldegphi/alldegphi.10000-19999">10000-19999</option>
<option value="alldegphi/alldegphi.20000-29999">20000-29999</option>
<option value="alldegphi/alldegphi.30000-39999">30000-39999</option>
<option value="alldegphi/alldegphi.40000-49999">40000-49999</option>
<option value="alldegphi/alldegphi.50000-59999">50000-59999</option>
<option value="alldegphi/alldegphi.60000-69999">60000-69999</option>
<option value="alldegphi/alldegphi.70000-79999">70000-79999</option>
<option value="alldegphi/alldegphi.80000-89999">80000-89999</option>
<option value="alldegphi/alldegphi.90000-99999">90000-99999</option>
<option value="alldegphi/alldegphi.100000-109999">100000-109999</option>
<option value="alldegphi/alldegphi.110000-119999">110000-119999</option>
<option value="alldegphi/alldegphi.120000-129999">120000-129999</option>
<option value="alldegphi/alldegphi.130000-139999">130000-139999</option>
<option value="alldegphi/alldegphi.140000-149999">140000-149999</option>
<option value="alldegphi/alldegphi.150000-159999">150000-159999</option>
<option value="alldegphi/alldegphi.160000-169999">160000-169999</option>
<option value="alldegphi/alldegphi.170000-179999">170000-179999</option>
<option value="alldegphi/alldegphi.180000-189999">180000-189999</option>
<option value="alldegphi/alldegphi.190000-199999">190000-199999</option>
<option value="alldegphi/alldegphi.200000-209999">200000-209999</option>
<option value="alldegphi/alldegphi.210000-219999">210000-219999</option>
<option value="alldegphi/alldegphi.220000-229999">220000-229999</option>
<option value="alldegphi/alldegphi.230000-239999">230000-239999</option>
<option value="alldegphi/alldegphi.240000-249999">240000-249999</option>
<option value="alldegphi/alldegphi.250000-259999">250000-259999</option>
<option value="alldegphi/alldegphi.260000-269999">260000-269999</option>
<option value="alldegphi/alldegphi.270000-279999">270000-279999</option>
<option value="alldegphi/alldegphi.280000-289999">280000-289999</option>
<option value="alldegphi/alldegphi.290000-299999">290000-299999</option>
<option value="alldegphi/alldegphi.300000-309999">300000-309999</option>
<option value="alldegphi/alldegphi.310000-319999">310000-319999</option>
<option value="alldegphi/alldegphi.320000-329999">320000-329999</option>
<option value="alldegphi/alldegphi.330000-339999">330000-339999</option>
<option value="alldegphi/alldegphi.340000-349999">340000-349999</option>
<option value="alldegphi/alldegphi.350000-359999">350000-359999</option>
<option value="alldegphi/alldegphi.360000-369999">360000-369999</option>
<option value="alldegphi/alldegphi.370000-379999">370000-379999</option>
<option value="alldegphi/alldegphi.380000-389999">380000-389999</option>
<option value="alldegphi/alldegphi.390000-399999">390000-399999</option>
<option value="alldegphi/alldegphi.400000-409999">400000-409999</option>
<option value="alldegphi/alldegphi.410000-419999">410000-419999</option>
<option value="alldegphi/alldegphi.420000-429999">420000-429999</option>
<option value="alldegphi/alldegphi.430000-439999">430000-439999</option>
<option value="alldegphi/alldegphi.440000-449999">440000-449999</option>
<option value="alldegphi/alldegphi.450000-459999">450000-459999</option>
<option value="alldegphi/alldegphi.460000-469999">460000-469999</option>
<option value="alldegphi/alldegphi.470000-479999">470000-479999</option>
<option value="alldegphi/alldegphi.480000-489999">480000-489999</option>
<option value="alldegphi/alldegphi.490000-499999">490000-499999</option>
</select>
<input type=button value="Fetch" onClick="JumpToIt(this.form)">
</form>
A table of the degree of the modular parametrizations of every curve.
<br>
Data format with sample line:
<table cellspacing="10" border="1">
<tbody><tr>
<th>N</th> <th>id</th> <th>#</th> <th>curve</th> <th>degree</th>
</tr>
<tr>
<td>11</td> <td>a</td> <td>1</td> <td>[0,-1,1,-10,-20]</td> <td> 1</td>
</tr>
<tr>
<td>11</td> <td>a</td> <td>2</td> <td>[0,-1,1,-7820,-263580]</td> <td> 5</td>
</tr>
<tr>
<td>11</td> <td>a</td> <td>3</td> <td>[0,-1,1,0,0]</td> <td> 5</td>
</tr></tbody></table>
</li>
</ul>
<h3>
TABLE SIX: ISOGENY MATRICES
</h3>
<ul>
<li>
<form>
<select name="url" width=30>
<option value="None">Select a conductor range</option>
<option value="allisog/allisog.00000-09999">1-9999</option>
<option value="allisog/allisog.10000-19999">10000-19999</option>
<option value="allisog/allisog.20000-29999">20000-29999</option>
<option value="allisog/allisog.30000-39999">30000-39999</option>
<option value="allisog/allisog.40000-49999">40000-49999</option>
<option value="allisog/allisog.50000-59999">50000-59999</option>
<option value="allisog/allisog.60000-69999">60000-69999</option>
<option value="allisog/allisog.70000-79999">70000-79999</option>
<option value="allisog/allisog.80000-89999">80000-89999</option>
<option value="allisog/allisog.90000-99999">90000-99999</option>
<option value="allisog/allisog.100000-109999">100000-109999</option>
<option value="allisog/allisog.110000-119999">110000-119999</option>
<option value="allisog/allisog.120000-129999">120000-129999</option>
<option value="allisog/allisog.130000-139999">130000-139999</option>
<option value="allisog/allisog.140000-149999">140000-149999</option>
<option value="allisog/allisog.150000-159999">150000-159999</option>
<option value="allisog/allisog.160000-169999">160000-169999</option>
<option value="allisog/allisog.170000-179999">170000-179999</option>
<option value="allisog/allisog.180000-189999">180000-189999</option>
<option value="allisog/allisog.190000-199999">190000-199999</option>
<option value="allisog/allisog.200000-209999">200000-209999</option>
<option value="allisog/allisog.210000-219999">210000-219999</option>
<option value="allisog/allisog.220000-229999">220000-229999</option>
<option value="allisog/allisog.240000-249999">240000-249999</option>
<option value="allisog/allisog.250000-259999">250000-259999</option>
<option value="allisog/allisog.260000-269999">260000-269999</option>
<option value="allisog/allisog.270000-279999">270000-279999</option>
<option value="allisog/allisog.280000-289999">280000-289999</option>
<option value="allisog/allisog.290000-299999">290000-299999</option>
<option value="allisog/allisog.300000-309999">300000-309999</option>
<option value="allisog/allisog.310000-319999">310000-319999</option>
<option value="allisog/allisog.320000-329999">320000-329999</option>
<option value="allisog/allisog.330000-339999">330000-339999</option>
<option value="allisog/allisog.340000-349999">340000-349999</option>
<option value="allisog/allisog.350000-359999">350000-359999</option>
<option value="allisog/allisog.360000-369999">360000-369999</option>
<option value="allisog/allisog.370000-379999">370000-379999</option>
<option value="allisog/allisog.380000-389999">380000-389999</option>
<option value="allisog/allisog.390000-399999">390000-399999</option>
<option value="allisog/allisog.400000-409999">400000-409999</option>
<option value="allisog/allisog.410000-419999">410000-419999</option>
<option value="allisog/allisog.420000-429999">420000-429999</option>
<option value="allisog/allisog.430000-439999">430000-439999</option>
<option value="allisog/allisog.440000-449999">440000-449999</option>
<option value="allisog/allisog.450000-459999">450000-459999</option>
<option value="allisog/allisog.460000-469999">460000-469999</option>
<option value="allisog/allisog.470000-479999">470000-479999</option>
<option value="allisog/allisog.480000-489999">480000-489999</option>
<option value="allisog/allisog.490000-499999">490000-499999</option>
</select>
<input type=button value="Fetch" onClick="JumpToIt(this.form)">
</form>
A table giving the degrees of isogenies within each isogeny class. One row
for each isogeny class.
<br>
Data format with sample line:
<table cellspacing="10" border="1">
<tbody><tr>
<th>N</th> <th>class</th> <th>#</th> <th>[a1,a2,a3,a4,a6]</th>
<th>curves in the class</th>
<th>isogeny matrix</th>
</tr>
<tr>
<td>14</td> <td>a</td> <td>1</td> <td>[1,0,1,4,-6]</td>
<td>[[1,0,1,4,-6],[1,0,1,-36,-70],[1,0,1,-171,-874],[1,0,1,-1,0],[1,0,1,-2731,-55146],[1,0,1,-11,12]]</td>
<td>[[1,2,3,3,6,6],[2,1,6,6,3,3],[3,6,1,9,2,18],[3,6,9,1,18,2],[6,3,2,18,1,9],[6,3,18,2,9,1]]</td>
</tr></tbody></table>
where the isogeny matrix has (<i>i</i>,<i>j</i>) entry <i>d</i> when
the there is a cyclic isogeny of degree <i>d</i> from curve <i>i</i>
to curve <i>j</i>.
</li></ul>
<h3>
TABLE SEVEN: INTEGRAL POINTS
</h3>
<ul>
<li>
<form>
<select name="url" width=30>
<option value="None">Select a conductor range</option>
<option value="intpts/intpts.00000-09999">1-9999</option>
<option value="intpts/intpts.10000-19999">10000-19999</option>
<option value="intpts/intpts.20000-29999">20000-29999</option>
<option value="intpts/intpts.30000-39999">30000-39999</option>
<option value="intpts/intpts.40000-49999">40000-49999</option>
<option value="intpts/intpts.50000-59999">50000-59999</option>
<option value="intpts/intpts.60000-69999">60000-69999</option>
<option value="intpts/intpts.70000-79999">70000-79999</option>
<option value="intpts/intpts.80000-89999">80000-89999</option>
<option value="intpts/intpts.90000-99999">90000-99999</option>
<option value="intpts/intpts.100000-109999">100000-109999</option>
<option value="intpts/intpts.110000-119999">110000-119999</option>
<option value="intpts/intpts.120000-129999">120000-129999</option>
<option value="intpts/intpts.130000-139999">130000-139999</option>
<option value="intpts/intpts.140000-149999">140000-149999</option>
<option value="intpts/intpts.150000-159999">150000-159999</option>
<option value="intpts/intpts.160000-169999">160000-169999</option>
<option value="intpts/intpts.170000-179999">170000-179999</option>
<option value="intpts/intpts.180000-189999">180000-189999</option>
<option value="intpts/intpts.190000-199999">190000-199999</option>
<option value="intpts/intpts.200000-209999">200000-209999</option>
<option value="intpts/intpts.210000-219999">210000-219999</option>
<option value="intpts/intpts.220000-229999">220000-229999</option>
<option value="intpts/intpts.230000-239999">230000-239999</option>
<option value="intpts/intpts.240000-249999">240000-249999</option>
<option value="intpts/intpts.250000-259999">250000-259999</option>
<option value="intpts/intpts.260000-269999">260000-269999</option>
<option value="intpts/intpts.270000-279999">270000-279999</option>
<option value="intpts/intpts.280000-289999">280000-289999</option>
<option value="intpts/intpts.290000-299999">290000-299999</option>
<option value="intpts/intpts.300000-309999">300000-309999</option>
<option value="intpts/intpts.310000-319999">310000-319999</option>
<option value="intpts/intpts.320000-329999">320000-329999</option>
<option value="intpts/intpts.330000-339999">330000-339999</option>
<option value="intpts/intpts.340000-349999">340000-349999</option>
<option value="intpts/intpts.350000-359999">350000-359999</option>
<option value="intpts/intpts.360000-369999">360000-369999</option>
<option value="intpts/intpts.370000-379999">370000-379999</option>
<option value="intpts/intpts.380000-389999">380000-389999</option>
<option value="intpts/intpts.390000-399999">390000-399999</option>
<option value="intpts/intpts.400000-409999">400000-409999</option>
<option value="intpts/intpts.410000-419999">410000-419999</option>
<option value="intpts/intpts.420000-429999">420000-429999</option>
<option value="intpts/intpts.430000-439999">430000-439999</option>
<option value="intpts/intpts.440000-449999">440000-449999</option>
<option value="intpts/intpts.450000-459999">450000-459999</option>
<option value="intpts/intpts.460000-469999">460000-469999</option>
<option value="intpts/intpts.470000-479999">470000-479999</option>
<option value="intpts/intpts.480000-489999">480000-489999</option>
<option value="intpts/intpts.490000-499999">490000-499999</option>
</select>
<input type=button value="Fetch" onClick="JumpToIt(this.form)">
</form>
A table giving the x-coordinates of all integral points on all curves.
<br>
Data format with sample line:
<table cellspacing="10" border="1">
<tbody><tr>
<th>Curve</th> <th>[a1,a2,a3,a4,a6]</th>
<th align="left">x-coordinates of integral points</th>
</tr>
<tr>
<td>114114bz1</td>
<td>[1,0,0,-858375,380956041]</td>
<td>[-1098,-1042,-990,-954,-756,-522,-426,-72,36,102,270,354,414,498,596,630,726,918,960,1334,1590,1818,1974,2702,3006,3690,5250,6966,8352,9702,18054,24438,31848,48150,119988,295254,913014]</td>
</tr></tbody></table>
</li></ul>
<h3>
<a NAME="optimality">TABLE EIGHT: OPTIMALITY AND THE MANIN CONSTANT</a>
</h3>
<ul>
<li>
<p>
For isogeny classes of curves of conductor greater than 400000, we have
not yet determined in all cases which curve in each class is optimal.
However, in all cases we have verified that the Manin constant of the
optimal curve is equal to 1 (as it is conjectured to be for every
optimal curve), even in cases where we do not know for sure which
curve is optimal.
</p>
<p>
While we can (using our modular symbols programs) determine the
optimal curve in any individual case, this takes a long time to do for
all remaining cases; this is ongoing. For more details on this, see
my Appendix to the paper "The Manin Constant" by Amod Agashe, Ken
Ribet and William Stein [Pure and Applied Mathematics Quarterly,
Vol. 2 no.2 (2006), pp. 617-636.]
and <a href="manin.txt">these
detailed notes</a> with full results for all conductors to 500000.
These updated results include the proof that Manin's constant is 1 in
all cases, together with a list of which curves in the class might be
optimal, given the incomplete modular symbol computations carried out
to date. Note, however, that it follows from computation of the
modular degrees of all curves in the class (which computation is
conditional on Stevens's conjecture) that the optimal curve is always
the first curve listed.
</p>
<p>
<form>
<select name="url" width=30>
<option value="None">Select a conductor range</option>
<option value="opt_man/opt_man.00000-09999">00000-09999</option>
<option value="opt_man/opt_man.10000-19999">10000-19999</option>
<option value="opt_man/opt_man.20000-29999">20000-29999</option>
<option value="opt_man/opt_man.30000-39999">30000-39999</option>
<option value="opt_man/opt_man.40000-49999">40000-49999</option>
<option value="opt_man/opt_man.50000-59999">50000-59999</option>
<option value="opt_man/opt_man.60000-69999">60000-69999</option>
<option value="opt_man/opt_man.70000-79999">70000-79999</option>
<option value="opt_man/opt_man.80000-89999">80000-89999</option>
<option value="opt_man/opt_man.90000-99999">90000-99999</option>
<option value="opt_man/opt_man.100000-109999">100000-109999</option>
<option value="opt_man/opt_man.110000-119999">110000-119999</option>
<option value="opt_man/opt_man.120000-129999">120000-129999</option>
<option value="opt_man/opt_man.130000-139999">130000-139999</option>
<option value="opt_man/opt_man.140000-149999">140000-149999</option>
<option value="opt_man/opt_man.150000-159999">150000-159999</option>
<option value="opt_man/opt_man.160000-169999">160000-169999</option>
<option value="opt_man/opt_man.170000-179999">170000-179999</option>
<option value="opt_man/opt_man.180000-189999">180000-189999</option>
<option value="opt_man/opt_man.190000-199999">190000-199999</option>
<option value="opt_man/opt_man.200000-209999">200000-209999</option>
<option value="opt_man/opt_man.210000-219999">210000-219999</option>
<option value="opt_man/opt_man.220000-229999">220000-229999</option>
<option value="opt_man/opt_man.230000-239999">230000-239999</option>
<option value="opt_man/opt_man.240000-249999">240000-249999</option>
<option value="opt_man/opt_man.250000-259999">250000-259999</option>
<option value="opt_man/opt_man.260000-269999">260000-269999</option>
<option value="opt_man/opt_man.270000-279999">270000-279999</option>
<option value="opt_man/opt_man.280000-289999">280000-289999</option>
<option value="opt_man/opt_man.290000-299999">290000-299999</option>
<option value="opt_man/opt_man.300000-309999">300000-309999</option>
<option value="opt_man/opt_man.310000-319999">310000-319999</option>
<option value="opt_man/opt_man.320000-329999">320000-329999</option>
<option value="opt_man/opt_man.330000-339999">330000-339999</option>
<option value="opt_man/opt_man.340000-349999">340000-349999</option>
<option value="opt_man/opt_man.350000-359999">350000-359999</option>
<option value="opt_man/opt_man.360000-369999">360000-369999</option>
<option value="opt_man/opt_man.370000-379999">370000-379999</option>
<option value="opt_man/opt_man.380000-389999">380000-389999</option>
<option value="opt_man/opt_man.390000-399999">390000-399999</option>
<option value="opt_man/opt_man.400000-409999">400000-409999</option>
<option value="opt_man/opt_man.410000-419999">410000-419999</option>
<option value="opt_man/opt_man.420000-429999">420000-429999</option>
<option value="opt_man/opt_man.430000-439999">430000-439999</option>
<option value="opt_man/opt_man.440000-449999">440000-449999</option>
<option value="opt_man/opt_man.450000-459999">450000-459999</option>
<option value="opt_man/opt_man.460000-469999">460000-469999</option>
<option value="opt_man/opt_man.470000-479999">470000-479999</option>
<option value="opt_man/opt_man.480000-489999">480000-489999</option>
<option value="opt_man/opt_man.490000-499999">490000-499999</option>
</select>
<input type=button value="Fetch" onClick="JumpToIt(this.form)">
</form>
Table of results known regarding optimality and Manin constant in all
isogeny classes. For conductors greater than 400000, the values of
the Manin constant are conditional on the first curve in the class
being optimal.
<br>
Data format with sample lines:
<table cellspacing="10" border="1">
<tbody><tr>
<th>N</th> <th>class</th> <th>#</th> <th>[a1,a2,a3,a4,a6]</th> <th>Optimality code</th> <th>Manin constant</th>
</tr>
<tr><td>11</td><td>a</td><td>1</td><td>[0,-1,1,-10,-20]</td><td>1</td><td>1</td></tr>
<tr><td>11</td><td>a</td><td>2</td><td>[0,-1,1,-7820,-263580]</td><td>0</td><td>1</td></tr>
<tr><td>11</td><td>a</td><td>3</td><td>[0,-1,1,0,0]</td><td>0</td><td>5</td></tr>
<tr><td>499992</td><td>a</td><td>1</td><td>[0,-1,0,4481,148204]</td><td>3</td><td>1</td></tr>
<tr><td>499992</td><td>a</td><td>2</td><td>[0,-1,0,-29964,1526004]</td><td>3</td><td>1</td></tr>
<tr><td>499992</td><td>a</td><td>3</td><td>[0,-1,0,-446624,115024188]</td><td>3</td><td>1</td></tr>
<tr><td>499992</td><td>a</td><td>4</td><td>[0,-1,0,-164424,-24344100]</td><td>0</td><td>1</tr>
</tbody></table>
</p>
<p>
The optimality code is 0 for "not optimal", 1 for "optimal"
and <i>n</i> for "one of <i>n</i> possibly optimal curves in this
isogeny class". In the case of isogeny class 11a above, out of the
three curves in the class, the optimal curve is 11a1 (which is
$X_0(11)$), the first two curves have Manin constant 1, while the
curve 11a3 (which is $X_1(11)$) has Manin constant equal to 5. In the
class 499992a, out of four curves, the optimal curve is certainly one
of the first 3; and if the optimal curve is indeed 499992a1 then all
the Manin constants are equal to 1.
</p>
</li>
</ul>
<h3>
TABLE NINE: IMAGES OF GALOIS REPRESENTATIONS
</h3>
<ul>
<li>
<form>
<select name="url" width=30>
<option value="None">Select a conductor range</option>
<option value="galrep/galrep.00000-09999">1-9999</option>
<option value="galrep/galrep.10000-19999">10000-19999</option>
<option value="galrep/galrep.20000-29999">20000-29999</option>
<option value="galrep/galrep.30000-39999">30000-39999</option>
<option value="galrep/galrep.40000-49999">40000-49999</option>
<option value="galrep/galrep.50000-59999">50000-59999</option>
<option value="galrep/galrep.60000-69999">60000-69999</option>
<option value="galrep/galrep.70000-79999">70000-79999</option>
<option value="galrep/galrep.80000-89999">80000-89999</option>
<option value="galrep/galrep.90000-99999">90000-99999</option>
<option value="galrep/galrep.100000-109999">100000-109999</option>
<option value="galrep/galrep.110000-119999">110000-119999</option>
<option value="galrep/galrep.120000-129999">120000-129999</option>
<option value="galrep/galrep.130000-139999">130000-139999</option>
<option value="galrep/galrep.140000-149999">140000-149999</option>
<option value="galrep/galrep.150000-159999">150000-159999</option>
<option value="galrep/galrep.160000-169999">160000-169999</option>
<option value="galrep/galrep.170000-179999">170000-179999</option>
<option value="galrep/galrep.180000-189999">180000-189999</option>
<option value="galrep/galrep.190000-199999">190000-199999</option>
<option value="galrep/galrep.200000-209999">200000-209999</option>
<option value="galrep/galrep.210000-219999">210000-219999</option>
<option value="galrep/galrep.220000-229999">220000-229999</option>
<option value="galrep/galrep.230000-239999">230000-239999</option>
<option value="galrep/galrep.240000-249999">240000-249999</option>
<option value="galrep/galrep.250000-259999">250000-259999</option>
<option value="galrep/galrep.260000-269999">260000-269999</option>
<option value="galrep/galrep.270000-279999">270000-279999</option>
<option value="galrep/galrep.280000-289999">280000-289999</option>
<option value="galrep/galrep.290000-299999">290000-299999</option>
<option value="galrep/galrep.300000-309999">300000-309999</option>
<option value="galrep/galrep.310000-319999">310000-319999</option>
<option value="galrep/galrep.320000-329999">320000-329999</option>
<option value="galrep/galrep.330000-339999">330000-339999</option>
<option value="galrep/galrep.340000-349999">340000-349999</option>
<option value="galrep/galrep.350000-359999">350000-359999</option>
<option value="galrep/galrep.360000-369999">360000-369999</option>
<option value="galrep/galrep.370000-379999">370000-379999</option>
<option value="galrep/galrep.380000-389999">380000-389999</option>
<option value="galrep/galrep.390000-399999">390000-399999</option>
<option value="galrep/galrep.400000-409999">400000-409999</option>
<option value="galrep/galrep.410000-419999">410000-419999</option>
<option value="galrep/galrep.420000-429999">420000-429999</option>
<option value="galrep/galrep.430000-439999">430000-439999</option>
<option value="galrep/galrep.440000-449999">440000-449999</option>
<option value="galrep/galrep.450000-459999">450000-459999</option>
<option value="galrep/galrep.460000-469999">460000-469999</option>
<option value="galrep/galrep.470000-479999">470000-479999</option>
<option value="galrep/galrep.480000-489999">480000-489999</option>
<option value="galrep/galrep.490000-499999">490000-499999</option>
</select>
<input type=button value="Fetch" onClick="JumpToIt(this.form)">
</form>
A table giving the for each elliptic curve the primes <i>p</i> for
which the mod-<i>p</i> Galois representation is not maximal, as
computed by Andrew Sutherland, together with a code identifying the
image (as a subgroup of GL(2,<i>p</i>), up to conjugation). For
curves with CM the representation is never surjective and the image is
shown when it is not maximal, where maximal means as large as possible
given the constraints imposed by the endomorphism ring of E. For
curves without CM, the representation is surjective for all but
finitely many primes (Serre) and is conjectured to be surjective
for <i>p</i>>37, but this was only checked for 37<<i>p</i><80.
<br>
Data format with sample lines:
<table cellspacing="10" border="1">
<tbody><tr>
<th>Curve</th><th align="left">list of non-surjective images</th>
</tr>
<tr>
<td>11a1</td><td>5Cs.1.1</td>
</tr>
<tr>
<td>27a1</td><td>3Cs.1.1</td>
</tr>
<tr>
<td>37a1</td><td> </td>
</tr>
</tbody></table>
</li></ul>
<h3>
TABLE TEN: IMAGES OF TWO-ADIC GALOIS REPRESENTATION
</h3>
<ul>
<li>
<form>
<select name="url" width=30>
<option value="None">Select a conductor range</option>
<option value="2adic/2adic.00000-09999">1-9999</option>
<option value="2adic/2adic.10000-19999">10000-19999</option>
<option value="2adic/2adic.20000-29999">20000-29999</option>
<option value="2adic/2adic.30000-39999">30000-39999</option>
<option value="2adic/2adic.40000-49999">40000-49999</option>
<option value="2adic/2adic.50000-59999">50000-59999</option>
<option value="2adic/2adic.60000-69999">60000-69999</option>
<option value="2adic/2adic.70000-79999">70000-79999</option>
<option value="2adic/2adic.80000-89999">80000-89999</option>
<option value="2adic/2adic.90000-99999">90000-99999</option>
<option value="2adic/2adic.100000-109999">100000-109999</option>
<option value="2adic/2adic.110000-119999">110000-119999</option>
<option value="2adic/2adic.120000-129999">120000-129999</option>
<option value="2adic/2adic.130000-139999">130000-139999</option>
<option value="2adic/2adic.140000-149999">140000-149999</option>
<option value="2adic/2adic.150000-159999">150000-159999</option>
<option value="2adic/2adic.160000-169999">160000-169999</option>
<option value="2adic/2adic.170000-179999">170000-179999</option>
<option value="2adic/2adic.180000-189999">180000-189999</option>
<option value="2adic/2adic.190000-199999">190000-199999</option>
<option value="2adic/2adic.200000-209999">200000-209999</option>
<option value="2adic/2adic.210000-219999">210000-219999</option>
<option value="2adic/2adic.220000-229999">220000-229999</option>
<option value="2adic/2adic.230000-239999">230000-239999</option>
<option value="2adic/2adic.240000-249999">240000-249999</option>
<option value="2adic/2adic.250000-259999">250000-259999</option>
<option value="2adic/2adic.260000-269999">260000-269999</option>
<option value="2adic/2adic.270000-279999">270000-279999</option>
<option value="2adic/2adic.280000-289999">280000-289999</option>
<option value="2adic/2adic.290000-299999">290000-299999</option>
<option value="2adic/2adic.300000-309999">300000-309999</option>
<option value="2adic/2adic.310000-319999">310000-319999</option>
<option value="2adic/2adic.320000-329999">320000-329999</option>
<option value="2adic/2adic.330000-339999">330000-339999</option>
<option value="2adic/2adic.340000-349999">340000-349999</option>
<option value="2adic/2adic.350000-359999">350000-359999</option>
<option value="2adic/2adic.360000-369999">360000-369999</option>
<option value="2adic/2adic.370000-379999">370000-379999</option>
<option value="2adic/2adic.380000-389999">380000-389999</option>
<option value="2adic/2adic.390000-399999">390000-399999</option>
<option value="2adic/2adic.400000-409999">400000-409999</option>
<option value="2adic/2adic.410000-419999">410000-419999</option>
<option value="2adic/2adic.420000-429999">420000-429999</option>
<option value="2adic/2adic.430000-439999">430000-439999</option>
<option value="2adic/2adic.440000-449999">440000-449999</option>
<option value="2adic/2adic.450000-459999">450000-459999</option>
<option value="2adic/2adic.460000-469999">460000-469999</option>
<option value="2adic/2adic.470000-479999">470000-479999</option>
<option value="2adic/2adic.480000-489999">480000-489999</option>
<option value="2adic/2adic.490000-499999">490000-499999</option>
</select>
<input type=button value="Fetch" onClick="JumpToIt(this.form)">
</form>
A table giving, for each elliptic curve without CM, the image of the
2-adic Galois representation, as computed using a Magma program
provided by Jeremy Rouse and David Zureick Brown.
There are 1208 possible images. The index is the index of the image
in GL(2,Z_2). The level is the largest n such that the image contains
the kernel of reduction modulo 2^n. The generators are generating
matrices for the image modulo the level. Note that the action of
Galois is on the right, so points in E[2^r] are represented as row
vectors v and if M is in GL_2(Z_2) the action of M is v |-> v*M.
The label is a label for the associated modular
curve. See <a href="http://users.wfu.edu/rouseja/2adic/index.html">Rouses's
web page</a> for details.
<br>
Data format with sample lines:
<table cellspacing="10" border="1">
<tbody><tr>
<th>N</th> <th>class</th> <th>#</th> <th>[a1,a2,a3,a4,a6]</th>
<th>index</th><th>level</th>
<th align="left">matrix generators</th>
<th>label</th>
</tr>
<tr>
<td>11</td></td><td>a</td><td>1</td><td>[0,-1,1,-10,-20]</td><td>1</td><td>1</td><td>[]</td><td>X1</td>
</tr>
<tr>
<td>15</td><td>a</td><td>1</td><td>[1,1,1,-10,-10]</td><td>96</td><td>8</td>
<td>[[5,4,2,3],[1,0,0,5],[1,4,0,5],[1,0,4,5]]</td><td>X187d</td>
</tr>
</tbody></table>
</li></ul>
<h3>
TABLE ELEVEN: TORSION GROWTH
</h3>
Torsion growth data has been computed so far for curves of conductor up
to 400000, for extensions of degree up to 23. (In degrees 11, 13,
17, 19, 22 and 23 there are no cases of torsion growth).
<ul>
<li>
<form>
<select name="deg" width=10>
<option value="None">Select a degree</option>
<option value="2">2</option>
<option value="3">3</option>
<option value="4">4</option>
<option value="5">5</option>
<option value="6">6</option>
<option value="7">7</option>
<option value="8">8</option>
<option value="9">9</option>
<option value="10">10</option>
<option value="12">12</option>
<option value="14">14</option>
<option value="15">15</option>
<option value="16">16</option>
<option value="18">18</option>
<option value="20">20</option>
<option value="21">21</option>
</select>
<select name="range" width=10>
<option value="None">Select a conductor range</option>
<option value="0-9999">0-9999</option>
<option value="10000-19999">10000-19999</option>
<option value="20000-29999">20000-29999</option>
<option value="30000-39999">30000-39999</option>
<option value="40000-49999">40000-49999</option>
<option value="50000-59999">50000-59999</option>
<option value="60000-69999">60000-69999</option>
<option value="70000-79999">70000-79999</option>
<option value="80000-89999">80000-89999</option>
<option value="90000-99999">90000-99999</option>
<option value="100000-109999">100000-109999</option>
<option value="110000-119999">110000-119999</option>
<option value="120000-129999">120000-129999</option>
<option value="130000-139999">130000-139999</option>
<option value="140000-149999">140000-149999</option>
<option value="150000-159999">150000-159999</option>
<option value="160000-169999">160000-169999</option>
<option value="170000-179999">170000-179999</option>
<option value="180000-189999">180000-189999</option>
<option value="190000-199999">190000-199999</option>
<option value="200000-209999">200000-209999</option>
<option value="210000-219999">210000-219999</option>
<option value="220000-229999">220000-229999</option>
<option value="230000-239999">230000-239999</option>
<option value="240000-249999">240000-249999</option>
<option value="250000-259999">250000-259999</option>
<option value="260000-269999">260000-269999</option>
<option value="270000-279999">270000-279999</option>
<option value="280000-289999">280000-289999</option>
<option value="290000-299999">290000-299999</option>
<option value="300000-309999">300000-309999</option>
<option value="310000-319999">310000-319999</option>
<option value="320000-329999">320000-329999</option>
<option value="330000-339999">330000-339999</option>
<option value="340000-349999">340000-349999</option>
<option value="350000-359999">350000-359999</option>
<option value="360000-369999">360000-369999</option>
<option value="370000-379999">370000-379999</option>
<option value="380000-389999">380000-389999</option>
<option value="390000-399999">390000-399999</option>
</select>
<input type=button value="Fetch" onClick="JumpToTorGro(this.form)">
</form>
<p>
For 2 ≤ <i>d</i> ≤ 23 we give for every curve <i>E</i> a list of the number
fields <i>K</i> of degree <i>d</i> (if any) such that <i>E</i>(<i>K</i>)<sub>tors</sub> is strictly
larger than <i>E</i>(<b>Q</b>)<sub>tors</sub> (and in case <i>d</i> is composite,
strictly larger than <i>E</i>(<i>K'</i>)<sub>tors</sub> for all subfields <i>K'</i>⊂
</i>K</i>).
</p>
<p>
Fields are specified by the coefficients of a canonical defining polynomial.
Torsion structure is shown as [n] or [m,n] with m dividing n.
</p>
<p>
Data format with sample lines (one in degree 2, one in degree 12):
<table cellspacing="10" border="1">
<tbody><tr>
<th>Curve label</th> <th>[Torsion][Field]</th> <th>[Torsion][Field]</th> <th>[Torsion][Field]</th>
</tr>
<tr>
<td>130014e1</td> <td>[2,2][1806,-1,1]</td> <td>[4][-23,-1,1]</td> <td>[4][175,-1,1]</td>
</tr>
<tr>
<td>130032e1</td> <td colspan=3>[4][-10346,8862,-13965,14728,-4689,-1362,387,156,129,-86,9,0,1]</td>
</tr>
<tr>
</tr>
</tbody></table>
</p>
<hr>
<p><b>Recent update
notes</b>: <a href="release_notes.md">27 November 2020</a>
</p><hr>
<tt>john dot cremona at gmail dot com</tt>
<hr>
</body>
</html>
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