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<h1><strong class="pkg">Congruence</strong></h1>


<h2>Congruence subgroups of <span class="SimpleMath">\(SL_2(ℤ)\)</span></h2>

<p>Version 1.2.7</p>

<p>28 August 2024</p>

</div>
<p><b>Ann Dooms
      
      
      
    </b>
<br />Email: <span class="URL"><a href="mailto:andooms@vub.ac.be">andooms@vub.ac.be</a></span>
<br />Homepage: <span class="URL"><a href="http://homepages.vub.ac.be/~andooms">http://homepages.vub.ac.be/~andooms</a></span>
<br />Address: <br />Department of Mathematics, Vrije Universiteit Brussel<br /> Pleinlaan 2, Brussels, B-1050 Belgium
</p><p><b>Eric Jespers
      
      
      
    </b>
<br />Email: <span class="URL"><a href="mailto:efjesper@vub.ac.be">efjesper@vub.ac.be</a></span>
<br />Homepage: <span class="URL"><a href="http://homepages.vub.ac.be/~efjesper">http://homepages.vub.ac.be/~efjesper</a></span>
<br />Address: <br />Department of Mathematics, Vrije Universiteit Brussel<br /> Pleinlaan 2, Brussels, B-1050 Belgium
</p><p><b>Olexandr Konovalov
      
      
       
    </b>
<br />Email: <span class="URL"><a href="mailto:obk1@st-andrews.ac.uk">obk1@st-andrews.ac.uk</a></span>
<br />Homepage: <span class="URL"><a href="https://olexandr-konovalov.github.io/">https://olexandr-konovalov.github.io/</a></span>
<br />Address: <br />School of Computer Science<br /> University of St Andrews<br /> Jack Cole Building, North Haugh,<br /> St Andrews, Fife, KY16 9SX, Scotland
</p><p><b>Helena Verrill
      
      
      
    </b>
<br />Email: <span class="URL"><a href="mailto:verrill@math.lsu.edu">verrill@math.lsu.edu</a></span>
<br />Homepage: <span class="URL"><a href="http://www.math.lsu.edu/~verrill/">http://www.math.lsu.edu/~verrill/</a></span>
<br />Address: <br />Department of Mathematics<br /> Louisiana State University<br /> Baton Rouge, Louisiana, 70803-4918<br /> USA
</p>

<p><a id="X7AA6C5737B711C89" name="X7AA6C5737B711C89"></a></p>
<h3>Abstract</h3>
<p>The <strong class="pkg">GAP</strong> package <strong class="pkg">Congruence</strong> provides functionality to work with congruence subgroups of <span class="SimpleMath">\(SL_2(ℤ)\)</span>.</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2006-2024 by Ann Dooms, Eric Jespers, Olexandr Konovalov and Helena Verrill.</p>

<p><strong class="pkg">Congruence</strong> is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. For details, see the FSF's own site <span class="URL"><a href="https://www.gnu.org/licenses/gpl.html">https://www.gnu.org/licenses/gpl.html</a></span>.</p>

<p>If you obtained <strong class="pkg">Congruence</strong>, we would be grateful for a short notification sent to one of the authors.</p>

<p>If you publish a result which was partially obtained with the usage of <strong class="pkg">Congruence</strong>, please cite it in the following form:</p>

<p>A. Dooms, E. Jespers, O. Konovalov and H. Verrill. <em>Congruence --- Congruence subgroups of <span class="SimpleMath">\(SL_2(ℤ)\)</span>, Version 1.2.7;</em> 2024 (<span class="URL"><a href="https://gap-packages.github.io/congruence/">https://gap-packages.github.io/congruence/</a></span>).</p>

<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>We are very grateful to Mong-Lung Lang, Chong-Hai Lim and Ser Peow Tan for their comments provided while implementing algorithms from <a href="chapBib_mj.html#biBLLT-Algorithm">[LLT95a]</a> and <a href="chapBib_mj.html#biBLLT-Hecke">[LLT95b]</a>, and to Francqui Stichting (Belgium) for the support of the third author.</p>

<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>

<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1_mj.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X80AE633F82C4D9BF">1.1 <span class="Heading">General aims of <strong class="pkg">Congruence</strong> package</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7DB566D5785B7DBC">1.2 <span class="Heading">Installation and system requirements</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap2_mj.html#X7B010EE67FACF45E">2 <span class="Heading">Construction of congruence subgroups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7B010EE67FACF45E">2.1 <span class="Heading">Construction of congruence subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7A61F693873F7136">2.1-1 PrincipalCongruenceSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7B8DB77B81BE58D7">2.1-2 CongruenceSubgroupGamma0</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7B4FBED17ECE2A7F">2.1-3 CongruenceSubgroupGammaUpper0</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7CFDC47279AC0E85">2.1-4 CongruenceSubgroupGamma1</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7C3FCDD878FE57ED">2.1-5 CongruenceSubgroupGammaUpper1</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7FE839377D7F45EB">2.1-6 IntersectionOfCongruenceSubgroups</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X8267F261874959E5">2.2 <span class="Heading">Properties of congruence subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X828F7E08787650DC">2.2-1 IsPrincipalCongruenceSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X85124A697E826AB4">2.2-2 IsCongruenceSubgroupGamma0</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7A03633C83A286F5">2.2-3 IsCongruenceSubgroupGammaUpper0</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X8262396080F3B0DD">2.2-4 IsCongruenceSubgroupGamma1</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7D731035834CF878">2.2-5 IsCongruenceSubgroupGammaUpper1</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X83B4E4FA7F4DFB97">2.2-6 IsIntersectionOfCongruenceSubgroups</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X8664A60E875EA5DE">2.3 <span class="Heading">Attributes of congruence subgroups</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7D5696F584970D21">2.3-1 LevelOfCongruenceSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X87302F8A7E44D67B">2.3-2 IndexInSL2Z</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7BF57D157824FFC8">2.3-3 DefiningCongruenceSubgroups</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7B15B49583DC9EF5">2.4 <span class="Heading">Operations for congruence subgroups</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X8146AC8587C65DEE">2.4-1 Random</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X87BDB89B7AAFE8AD"><code>2.4-2 \in</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7FC5BF527931FF4C">2.4-3 CanEasilyCompareCongruenceSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X79CA175481F8105F">2.4-4 IsSubset</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X83A0356F839C696F">2.4-5 Index</a></span>
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<div class="ContChap"><a href="chap3_mj.html#X85CABB30818CD99C">3 <span class="Heading">Farey symbols and their properties</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7B7B81E584CCA30C">3.1 <span class="Heading">Construction of Farey symbols</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F8F5919870A46FE">3.1-1 FareySymbolByData</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X845F9BA182F4E73B">3.1-2 IsValidFareySymbol</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X8016C45082AEC784">3.2 <span class="Heading">Properties of Farey symbols</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8245766978F02751">3.2-1 GeneralizedFareySequence</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X80BB58E58492D103">3.2-2 NumeratorOfGFSElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X87477604878BCD42">3.2-3 DenominatorOfGFSElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X83C941047D486000">3.2-4 LabelsOfFareySymbol</a></span>
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<div class="ContChap"><a href="chap4_mj.html#X831C60277F7D80B2">4 <span class="Heading">Farey symbols for congruence subgroups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7F43DB8B803F313F">4.1 <span class="Heading">Computation of the Farey symbol for a finite index subgroup</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8594896287DCFE8D">4.1-1 FareySymbol</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X80AE179D869BEE90">4.2 <span class="Heading">Computation of generators of a finite index subgroup from its Farey symbol</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8790C1498107A39A">4.2-1 MatrixByEvenInterval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X78779BDF7A1DB4AE">4.2-2 MatrixByOddInterval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7F792846795E3A63">4.2-3 MatrixByFreePairOfIntervals</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7905B050800E4416">4.2-4 GeneratorsByFareySymbol</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X79C44528864044C5">4.2-5 GeneratorsOfGroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7C5AB1D786207745">4.3 <span class="Heading">Other properties derived from Farey symbols</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X80EED34183408106">4.3-1 IndexInPSL2ZByFareySymbol</a></span>
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</div>
<div class="ContChap"><a href="chap5_mj.html#X82C56A367A418E7C">5 <span class="Heading">Service functions of the <strong class="pkg">Congruence</strong> package</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X86D04EE08437C320">5.1 <span class="Heading">Additional information displayed by <strong class="pkg">Congruence</strong> algorithms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X83B2A8607C2E6A38">5.1-1 InfoCongruence</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chapBib_mj.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd_mj.html"><span class="Heading">Index</span></a></div>
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