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Issue LMFDB#6276: display of elliptic curve generators
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@JohnCremona
JohnCremona committed Nov 28, 2024
1 parent a23a996 commit a2ebabc
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Showing 4 changed files with 83 additions and 89 deletions.
10 changes: 10 additions & 0 deletions lmfdb/ecnf/WebEllipticCurve.py
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Original file line number Diff line number Diff line change
Expand Up @@ -483,6 +483,13 @@ def make_E(self):
except AttributeError:
self.qc = "not determined"

# Mordell-Weil group
try:
invs = [0 for a in range(self.rank)] + list(self.torsion_structure)
self.mw_struct = "trivial" if len(invs) == 0 else r'\(' + r' \oplus '.join((r'\Z' if n == 0 else r'\Z/{%s}\Z' % n) for n in invs) + r'\)'
except AttributeError: # if self.rank not set
self.mw_struct = "unknown"

# Torsion
self.ntors = web_latex(self.torsion_order)
self.tr = len(self.torsion_structure)
Expand All @@ -494,6 +501,7 @@ def make_E(self):
self.tor_struct_pretty = r"\(\Z/%s\Z\oplus\Z/%s\Z\)" % tuple(self.torsion_structure)

self.torsion_gens = [web_point(parse_point(K,P)) for P in self.torsion_gens]
self.tor_gens_and_orders = list(zip(self.torsion_gens, self.torsion_structure))

# BSD data
#
Expand Down Expand Up @@ -568,8 +576,10 @@ def make_E(self):
# Generators
try:
self.gens = [web_point(parse_point(K, P)) for P in self.gens]
self.gens_and_heights = list(zip(self.gens,self.heights))
except AttributeError:
self.gens = []
self.gens_and_orders = []

# Global period
try:
Expand Down
112 changes: 32 additions & 80 deletions lmfdb/ecnf/templates/ecnf-curve.html
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Original file line number Diff line number Diff line change
Expand Up @@ -70,6 +70,37 @@ <h2>{{ KNOWL('ec.weierstrass_coeffs', title='Weierstrass equation') }}</h2>
{{ place_code('is_min') }}
</p>

<h2> {{ KNOWL('ec.mordell_weil_group', title='Mordell-Weil group') }} structure</h2>
<p>
{% if ec.rk == "not available" %}
Not computed ($ {{ec.rk_lb}} \le r \le {{ec.rk_ub}} $)
{% else %}
{{ ec.mw_struct}}
{%endif%}
</p>

{% if ec.ngens > 0 or ec.torsion_order > 1 %}

<h2> {{ KNOWL('ec.mw_generators', title="Mordell-Weil generators") }} </h2>
<div>
<p>
{% if ec.ngens < ec.rk_ub %}
{% if ec.ngens > 0 %} Only {{ec.ngens}} {% else %} No {% endif %} non-torsion
{% if ec.ngens == 1 %} generator is {% else %} generators are {% endif %} known.
{% endif %}
<table>
<tr><th>$P$</th><th>{{ KNOWL('ec.canonical_height', title="$\hat{h}(P)$")}}</th><th>Order</th></tr>
{% for P,h in ec.gens_and_heights %}
<tr><td>{{P}}</td><td>${{h}}$</td><td>$\infty$</td></tr>
{% endfor %}
{% for T,n in ec.tor_gens_and_orders %}
<tr><td>{{T}}</td><td>$0$</td><td>${{n}}$</td></tr>
{% endfor %}
</table>
</p>
</div>
{%endif%}


<h2>{{ KNOWL('ec.invariants', title='Invariants')}}</h2>

Expand Down Expand Up @@ -189,85 +220,6 @@ <h2>{{ KNOWL('ec.invariants', title='Invariants')}}</h2>

</table>

<h2> {{ KNOWL('ec.mordell_weil_group', title="Mordell-Weil group") }} </h2>
<div>
<p>
<table>
<tr>
<td align = 'left'>{{ KNOWL('ec.rank', title="Rank")}}:</td>
{% if ec.rk == "not available" %}
{% if ec.rank_bounds != "not available" %}
<td>\({{ ec.rk_lb }} \le r \le {{ec.rk_ub}}\)</td>
{% else %}
<td>$r$</td>
<td>&nbsp;</td>
<td>not available</td>
{% endif %}
{% else %}
<td>$r$</td>
<td>=</td>
<td>\({{ ec.rank }}\)</td>
{% endif %}
</tr>

{% if ec.ngens %}
<tr>
<td align = 'left'>
{% if ec.ngens==1 %}
{{KNOWL('ec.mw_generators','Non-torsion generator')}}:
{% else %}
{{KNOWL('ec.mw_generators','Non-torsion generators')}}:
{% endif %}
</td>
{% if ec.gens == 'not available' %}
<td>not available</td>
{% else %}
<td>$P$</td><td>=</td>
{% for gen in ec.gens %}
<td>{{ gen }}</td>
{% endfor %}
{% endif %}
</tr>
{% endif %}

{% if ec.heights %}
<tr>
<td align = 'left'>
{% if ec.ngens==1 %}
{{ KNOWL('ec.canonical_height', title="Height") }}:
{% else %}
{{ KNOWL('ec.canonical_height', title="Heights") }}:
{% endif %}
</td>
<td>$\hat{h}(P)$</td><td>&approx;</td>
{% for h in ec.heights %}
<td>\({{ h }}\)</td>
{% endfor %}
</tr>
{% endif %}

<tr>
<td align='left'>{{KNOWL('ec.torsion_subgroup','Torsion structure')}}:</td>
<td>$E(K)_{\mathrm{tor}}$</td>
<td>$\cong$</td>
<td>{{ ec.tor_struct_pretty }}</td>
</tr>
<tr><td colspan=2> {{ place_code('tors') }}</td></tr>
<!-- <tr><td colspan=2> {{ place_code('ntors') }}</td></tr> -->
{% if ec.tr %}
<tr>
<td align='left'>{% if ec.tr==1 %}{{KNOWL('ec.mw_generators','Torsion generator')}}{% else %}{{KNOWL('ec.mw_generators','Torsion generators')}}{% endif %}:</td>
<td>$T$</td>
<td>=</td>
{% for gen in ec.torsion_gens %}
<td>{{ gen }}</td>
{% endfor %}
</tr>
<tr><td colspan=2> {{ place_code('torgens') }}</td></tr>
{% endif %}
</table>
</p>
</div>

<h2> {{ KNOWL('ec.bsdconjecture', title='BSD invariants') }}</h2>
<div>
Expand Down Expand Up @@ -341,7 +293,7 @@ <h2> {{ KNOWL('ec.bsdconjecture', title='BSD invariants') }}</h2>

<tr>
<td align='left'>{{ KNOWL('ec.torsion_order', title='Torsion order') }}:</td>
<td>$E(K)_{\mathrm{tors}}$</td><td>=</td>
<td>$\#E(K)_{\mathrm{tor}}$</td><td>=</td>
<td>\({{ ec.torsion_order }}\)</td>
</tr>

Expand Down
42 changes: 35 additions & 7 deletions lmfdb/elliptic_curves/templates/ec-curve.html
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Original file line number Diff line number Diff line change
Expand Up @@ -102,6 +102,24 @@ <h2> {{ KNOWL('ec.mordell_weil_group', title='Mordell-Weil group') }} structure<

<p>{{ place_code('mwgroup') }}</p>

{% if data.mwbsd.rank or data.mwbsd.torsion_order %}
<h2> {{ KNOWL('ec.mw_generators', title="Mordell-Weil generators") }} </h2>
<div>
<p>
<table>
<tr><th>$P$</th><th>{{ KNOWL('ec.canonical_height', title="$\hat{h}(P)$")}}</th><th>Order</th></tr>
{% for P,h in data.mwbsd.gens_and_heights %}
<tr><td>${{P}}$</td><td>${{h}}$</td><td>$\infty$</td></tr>
{% endfor %}
{% for T,n in data.mwbsd.tor_gens_and_orders %}
<tr><td>${{T}}$</td><td>$0$</td><td>${{n}}$</td></tr>
{% endfor %}
</table>
</p>
</div>
{%endif%}

{#
{% if data.mwbsd.rank!=0 %}
{% if data.mwbsd.rank==1 %}
<h3> {{ KNOWL('ec.mw_generators', title='Non-torsion Mordell-Weil generator') }} and {{KNOWL('ec.canonical_height','height')}}</h3>
Expand Down Expand Up @@ -134,8 +152,9 @@ <h2> {{ KNOWL('ec.mw_generators', title='Torsion generators') }}</h2>
<p> {{ data.mwbsd.tor_gens |safe }} </p>
{{ place_code('tors') }}
{%endif %}
#}


{% if data.mwbsd.rank > 0 or data.mwbsd.torsion_order > 1 %}
<p><h2> {{ KNOWL('ec.q.integral_points', title='Integral points') }}</h2>
<div class="ip">
<p>
Expand All @@ -148,6 +167,7 @@ <h2> {{ KNOWL('ec.mw_generators', title='Torsion generators') }}</h2>
N.B. Only integral points which are combinations of known generators are shown.
</p>
{%endif %}
{%endif %}

<h2> Invariants </h2>
<table>
Expand Down Expand Up @@ -237,6 +257,13 @@ <h2> {{ KNOWL('ec.q.bsdconjecture', title='BSD invariants') }}</h2>
<td>{{ place_code('analytic_rank') }}</td>
</tr>

<tr>
<td>{{ KNOWL('ec.rank', title='Rank') }}:</td>
<td>$r$</td><td>&nbsp;=&nbsp;</td><td>$ {{ data.rank }}$
</td>
<td>{{ place_code('rank') }}</td>
</tr>

<tr>
<td>{{ KNOWL('ec.regulator', title='Regulator') }}:</td>
{% if data.mwbsd.reg == '?' %}
Expand Down Expand Up @@ -269,10 +296,16 @@ <h2> {{ KNOWL('ec.q.bsdconjecture', title='BSD invariants') }}</h2>

<tr>
<td>{{ KNOWL('ec.torsion_order', title='Torsion order') }}:</td>
<td>$E(\Q)_{\mathrm{tors}}$</td><td>&nbsp;=&nbsp;</td><td>${{ data.mwbsd.torsion }}$</td>
<td>$\#E(\Q)_{\mathrm{tor}}$</td><td>&nbsp;=&nbsp;</td><td>${{ data.mwbsd.torsion }}$</td>
<td>{{ place_code('ntors') }}</td>
</tr>

<tr>
<td>{{KNOWL('ec.q.special_value', title='Special value', special_value = data.special_value)}}:</td>
<td>$ {{ data.mwbsd.lder_name }}$</td><td>&nbsp;&approx;&nbsp;</td><td>${{ data.mwbsd.special_value }} $</td>
<td>{{ place_code('L1') }}</td>
</tr>

<tr>
<td>{{ KNOWL('ec.analytic_sha_order', title='Analytic order of &#1064;') }}:</td>
<td>{% if data.mwbsd.sha == '?' %}Not computed{% else %} &#1064;${}_{\mathrm{an}}$</td>
Expand All @@ -293,11 +326,6 @@ <h2> {{ KNOWL('ec.q.bsdconjecture', title='BSD invariants') }}</h2>
<td> ${{ data.mwbsd.regsha }}$</td>
</tr>
{% endif %}
<tr>
<td>{{KNOWL('ec.q.special_value', title='Special value', special_value = data.special_value)}}:</td>
<td>$ {{ data.mwbsd.lder_name }}$</td><td>&nbsp;&approx;&nbsp;</td><td>${{ data.mwbsd.special_value }} $</td>
<td>{{ place_code('L1') }}</td>
</tr>
</tbody>
</table>

Expand Down
8 changes: 6 additions & 2 deletions lmfdb/elliptic_curves/web_ec.py
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Original file line number Diff line number Diff line change
Expand Up @@ -657,9 +657,11 @@ def make_mwbsd(self):
mwbsd['int_points'] = "None"

# Generators (mod torsion) and heights:
mwbsd['generators'] = [raw_typeset(weighted_proj_to_affine_point(P)) for P in mwbsd['gens']] if mwbsd['ngens'] else ''
#mwbsd['generators'] = [raw_typeset(weighted_proj_to_affine_point(P)) for P in mwbsd['gens']] if mwbsd['ngens'] else []
mwbsd['generators'] = [weighted_proj_to_affine_point(P) for P in mwbsd['gens']] if mwbsd['ngens'] else []
mwbsd['heights'] = [RR(h) for h in mwbsd['heights']]

mwbsd['gens_and_heights'] = list(zip(mwbsd['generators'], mwbsd['heights']))

# Mordell-Weil group
invs = [0 for a in range(self.rank)] + list(self.torsion_structure)
mwbsd['mw_struct'] = "trivial" if len(invs) == 0 else r'\(' + r' \oplus '.join((r'\Z' if n == 0 else r'\Z/{%s}\Z' % n) for n in invs) + r'\)'
Expand All @@ -668,11 +670,13 @@ def make_mwbsd(self):
if mwbsd['torsion'] == 1:
mwbsd['tor_struct'] = ''
mwbsd['tor_gens'] = ''
mwbsd['tor_gens_and_orders'] = []
else:
mwbsd['tor_struct'] = r' \oplus '.join(r'\Z/{%s}\Z' % n for n in self.torsion_structure)
tor_gens_tmp = [weighted_proj_to_affine_point(P) for P in mwbsd['torsion_generators']]
mwbsd['tor_gens'] = raw_typeset(', '.join(str(P) for P in tor_gens_tmp),
', '.join(web_latex(P) for P in tor_gens_tmp))
mwbsd['tor_gens_and_orders'] = list(zip(tor_gens_tmp, self.torsion_structure))

# BSD invariants
if r >= 2:
Expand Down

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