ec-isoclass.html

{% extends 'homepage.html' %}
{% block content %}

{{ place_code('isogeny_class', is_top_snippet=True) }}

<style type="text/css">
#isogeny_class_table th, #isogeny_class_table td {
padding : 4px;
text-align: center;
}
</style>

<h2>{{KNOWL('ec.rank', title='Rank')}}</h2>
{{ place_code('rank') }}
<p>
{% if info.class_size==1 %} The elliptic curve {{info.curves[0].curve_label}} has
{% else %} The elliptic curves in class {{info.class_label}} have
{% endif %}
{{KNOWL('ec.rank', title='rank')}} \({{ info.rank}}\).
</p>

{% if info.has_lfunction %}
<h2>{{KNOWL('g2c.lfunction', title='L-function')}} data</h2>
<table>
    <tr><td valign=top> {{ KNOWL('g2c.bad_lfactors', title='Bad L-factors') }}:<td>
        <table id = "bad_lfactors_table">
        <tr>
            <th>Prime</th>
            <th align="left">L-Factor</th>
        </tr>
        {% for c in info.bad_lfactors_pretty %}
        <tr><td class="center">\({{ c[0] }}\)</a></td><td align="left">\({{ c[1] }}\)</td></tr>
        {% endfor %}
        </table>
    </tr>
    <tr><td>&nbsp;</td></tr>
    <tr><td valign=top> {{ KNOWL('g2c.good_lfactors', title='Good L-factors') }}:<td>
        <table id = "good_lfactors_table">
        <tr>
            <th>Prime</th>
            <th align="left">L-Factor</th>
            <th align="left">Isogeny Class over \(\mathbb{F}_p\)</th>
        </tr>
        {% for c in info.good_lfactors_pretty_with_label %}
        <tr>
            <td class="center">\({{ c[0] }}\)</a></td>
            <td align="left">\( {{ c[1] }}\)</td>
            <td>
              {% if c[3] == '' %}
                {{ c[2] }}
              {% else %}
                <a href="{{ c[3] }}">{{ c[2] }}</a>
              {% endif %}
            </td>
          </tr>
        {% endfor %}
        <tr><td class="center">$\cdots$</td><td>$\cdots$</td><td>$\cdots$</td>
        </table>
    </tr>
    <tr><td>&nbsp;</td></tr>
    <tr><td colspan="2">See <a href="{{info.lfunction_link}}">L-function page</a> for more information</td></tr>
</table>
{% endif %}

<h2>{{KNOWL('ec.complex_multiplication', title='Complex multiplication')}}</h2>
{% if info.cm %}
Each elliptic curve in class {{info.class_label}} has complex multiplication by an order in the imaginary quadratic field {{info.CMfield }}.
{% else %} The elliptic curves in class {{info.class_label}} do not have complex multiplication.
{% endif %}

<h2>
{{KNOWL('ec.q.modular_form', title='Modular form')}}
{% if info.newform_exists_in_db %}
<a href="{{info.newform_link}}">{{ info.newform_label }}</a>
{% else %}
{{ info.newform_label }}
{% endif %}
</h2>
{{ place_code('qexp') }}
<div>
    <form>
        <div class="output">
          <span id="modform_output">{{ info.newform | safe }}
          </span>
        </div>
        <div class="emptyspace"><br>
        </div>
        <button id="morebutton">Show more coefficients</button>
    </form>
</div>

<script type="text/javascript">
  var number_of_coefficients = 20;
  function more_handler(evt) {
      number_of_coefficients += number_of_coefficients;
      evt.preventDefault();
      $("#modform_output").load("{{info.modform_display}}"+number_of_coefficients,
          function() {
              {# render the output #}
              renderMathInElement($("#modform_output").get(0), katexOpts);
          });
  }
  $(function() {
      $("#morebutton").click(function(e) {more_handler(e)});
  });
  </script>
  
{% if info.class_size > 1 %}

<h2>{{ KNOWL('ec.isogeny_matrix',title='Isogeny matrix') }}</h2>

{{ place_code('isogeny_matrix') }}
<p>The \((i,j)\)-th entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the {{info.label_type}} numbering.
</p>
<p>
  \({{info.isogeny_matrix_str}}\)
</p>

{% endif %}

<h2> {{ KNOWL('ec.isogeny_graph', title='Isogeny graph') }} </h2>
{{ place_code('isogeny_graph') }}
{% if info.class_size > 1 %}
<p>The vertices are labeled with {{info.label_type}} labels, and the {{ KNOWL('ec.q.optimal', title='\( \Gamma_0(N) \)-optimal') }} curve is highlighted in blue.</p>
{% endif %}
<div id="isogeny-graph"></div>

{% set graph_layouts = info.graph_layouts %}
{% include "cytoscape_scripts.html" %}
<script>
  initIsogenyGraph('isogeny-graph', {{ info.graph_data | tojson }}, {{ info.graph_layouts | tojson }}, {{ info.graph_default_layout | tojson }});
</script>

<h2>Elliptic curves in class {{info.class_label}}</h2>
{{ place_code('curves') }}
<table id = "isogeny_class_table">
<tr>
<th>{{ KNOWL('ec.q.lmfdb_label', title='LMFDB label')}}</th>
{% if info.conductor < info.cremona_bound %}
<th>{{ KNOWL('ec.q.cremona_label', title='Cremona label')}}</th>
{% endif %}
<th>{{ KNOWL('ec.q.minimal_weierstrass_equation', title='Weierstrass coefficients') }}</th>
<th>{{ KNOWL('ec.j_invariant', title='j-invariant') }}</th>
<th>{{ KNOWL('ec.discriminant',  title='Discriminant') }}</th>
<th>{{ KNOWL('ec.torsion_subgroup', title='Torsion structure') }}</th>
<th>{{ KNOWL('ec.q.modular_degree', title='Modular degree') }}</th>
<th>{{ KNOWL('ec.q.faltings_height', title='Faltings height') }}</th>
{% if info.conductor < info.cremona_bound %}
<th>{{ KNOWL('ec.q.optimal', title='Optimality') }}</th>
{% endif %}
{#
<th>{{ KNOWL('ec.q.faltings_ratio', title='Faltings ratio') }}</th>
#}
{% if info.cm %}
<th>{{ KNOWL('ec.complex_multiplication', title='CM discriminant') }}</th>
{% endif %}
</tr>
{% for c in info.curves %}
{% if c.optimal %}
<tr bgcolor="{{color.ec_background}}">
{% else %}
<tr>
{% endif %}
<td class="center"><a href="{{c.curve_url_lmfdb}}">{{c.lmfdb_label}}</a></td>
{% if info.conductor < info.cremona_bound %}
<td class="center"><a href="{{c.curve_url_cremona}}">{{c.Clabel}}</a></td>
{% endif %}
<td class="center">\({{c.ai}}\)</td>
<td class="center">\({{c.j_inv}}\)</td>
<td class="center">\({{c.disc}}\)</td>
<td align="center">\({{c.torsion_structure}}\)</td>
<td align="center">
{% if c.degree==0 %}Not available{% else %}\({{c.degree}}\){% endif %}
</td>
<td align="center">
\({{c.FH}}\)
</td>
{% if info.conductor < info.cremona_bound %}
<td>
  {% if c.optimal %}
\(\Gamma_0(N)\)-optimal{% if not info.optimality_known %}<sup>*</sup>{% endif %}
{#
  {% if c.faltings_index==0 %}
, \(\Gamma_1(N)\)-optimal{% if info.class_size>1 %}<sup>**</sup>{% endif %}
  {% endif %}
#}
{% else %}
{#
{% if c.faltings_index==0 %}
\(\Gamma_1(N)\)-optimal{% if info.class_size>1 %}<sup>**</sup>{% endif %}
{% else %}
#}
&nbsp;
{#
{% endif %}
#}
{% endif %}
</td>
{% endif %}
{#
<td>\({{c.faltings_ratio}}\)</td>
#}
{% if info.cm %}
<td>\({{c.cm}}\)</td>
{% endif %}
</tr>
{% endfor %}
</table>

{% if info.conductor < info.cremona_bound %}
{% if not info.optimality_known %}<sup>*</sup>optimality has not been
determined rigorously for conductors over {{ info.optimality_bound }}.  In
this case the optimal curve is certainly one of the {{
info.curves[0].optimality }} curves highlighted, and conditionally
curve {{info.optimal_label}}.{% endif %}
{% endif %}

{#
{% if info.class_size>1 %}<sup>**</sup>conjecturally; proved in some cases.{% endif %}
#}

{% endblock %}