From b64f1d2dab84727ecc441fa95dff68a223015ede Mon Sep 17 00:00:00 2001
From: John Cremona <john.cremona@gmail.com>
Date: Wed, 4 Dec 2024 04:28:43 -0500
Subject: [PATCH 01/11] EC/ECNF consistency in local data display

---
 lmfdb/ecnf/templates/ecnf-curve.html          | 30 ++++++++++++++-----
 lmfdb/elliptic_curves/templates/ec-curve.html |  8 ++---
 2 files changed, 26 insertions(+), 12 deletions(-)

diff --git a/lmfdb/ecnf/templates/ecnf-curve.html b/lmfdb/ecnf/templates/ecnf-curve.html
index baed8cc381..be4782efcf 100644
--- a/lmfdb/ecnf/templates/ecnf-curve.html
+++ b/lmfdb/ecnf/templates/ecnf-curve.html
@@ -276,7 +276,7 @@ <h2> {{ KNOWL('ec.bsdconjecture', title='BSD invariants') }}</h2>
         </tr>
 
         <tr>
-        <td align='left'>{{ KNOWL('ec.period', title='Period') }}:</td>
+        <td align='left'>{{ KNOWL('ec.period', title='Global period') }}:</td>
 	<td>$\Omega(E/K)$</td><td>&approx;</td>
         <td> {{ ec.omega }}</td>
         </tr>
@@ -341,25 +341,41 @@ <h2>{{KNOWL('ec.local_data', title='Local data')}} at {{KNOWL('ec.bad_reduction'
 <div>
  {{ place_code('localdata') }}
 
+<p>
+  This elliptic curve is {{ '' if ec.semistable else 'not' }} {{KNOWL('ec.semistable', title='semistable')}}.
+  There
+  {% if ec.n_bad_primes==0 %}
+  are no primes
+  {% else %}
+  {% if ec.n_bad_primes==1 %}
+  is only one prime $\frak{p}$
+  {% else %}
+  are {{ec.n_bad_primes }} primes $\frak{p}$
+  {% endif %}
+  {% endif %}
+  of {{KNOWL('ec.q.reduction_type', title='bad reduction')}}.
+
   {% if not ec.is_minimal %}
     Primes of good reduction for the curve but which divide the
     discriminant of the model above (if any) are included.
   {% endif %}
+</p>
+
 
 {% if ec.local_data %}
 <table class="ntdata"><thead>
 <tr>
-<th>prime</th>
-<th>Norm</th>
+<th>$\mathfrak{p}$</th>
+<th>$N(\mathfrak{p})$</th>
 <th>{{KNOWL('ec.tamagawa_number', title='Tamagawa number')}}</th>
 <th>{{KNOWL('ec.kodaira_symbol', title='Kodaira symbol')}}</th>
 <th>{{KNOWL('ec.reduction_type', title='Reduction type')}}</th>
 {% if ec.local_data.0.rootno %}
 <th>{{KNOWL('ec.local_root_number', title='Root number')}}</th>
 {% endif %}
-<th>{{KNOWL('ec.conductor_valuation', title='ord(\(\mathfrak{N}\))')}}</th>
-<th>{{KNOWL('ec.discriminant_valuation', title='ord(\(\mathfrak{D}\))')}}</th>
-<th>{{KNOWL('ec.j_invariant_denominator_valuation', title='ord\((j)_{-}\)')}}</th>
+<th>{{KNOWL('ec.conductor_valuation', title='\(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\))')}}</th>
+<th>{{KNOWL('ec.discriminant_valuation', title='\(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\))')}}</th>
+<th>{{KNOWL('ec.j_invariant_denominator_valuation', title='\(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)')}}</th>
 </tr>
 </thead><tbody>
 {% for pr in ec.local_data %}
@@ -397,8 +413,6 @@ <h2>{{KNOWL('ec.local_data', title='Local data')}} at {{KNOWL('ec.bad_reduction'
 {% endfor %}
 </tbody>
 </table>
-{% else %}
-No primes of bad reduction.
 {% endif %}
 </div>
 
diff --git a/lmfdb/elliptic_curves/templates/ec-curve.html b/lmfdb/elliptic_curves/templates/ec-curve.html
index e49ec675c7..78c1c9fa1a 100644
--- a/lmfdb/elliptic_curves/templates/ec-curve.html
+++ b/lmfdb/elliptic_curves/templates/ec-curve.html
@@ -406,7 +406,7 @@ <h4> {{ KNOWL('ec.q.modular_degree', title='Modular degree') }}
 	{% endif %}
 	</p>
 
-    <h2> Local data </h2>
+<h2>{{KNOWL('ec.local_data', title='Local data')}} at {{KNOWL('ec.bad_reduction', title='primes of bad reduction')}} </h2>
 
 <style type="text/css">
 #local_data th, #local_data td, #serre_data th, #serre_data td {
@@ -433,9 +433,9 @@ <h2> Local data </h2>
 <th>{{KNOWL('ec.q.kodaira_symbol', title='Kodaira symbol')}}</th>
 <th>{{KNOWL('ec.q.reduction_type', title='Reduction type')}}</th>
 <th>{{KNOWL('ec.local_root_number', title='Root number')}}</th>
-<th>{{KNOWL('ec.conductor_valuation', title='$v_p(N)$')}}</th>
-<th>{{KNOWL('ec.discriminant_valuation', title='$v_p(\Delta)$')}}</th>
-<th>{{KNOWL('ec.j_invariant_denominator_valuation', title='$v_p(\mathrm{den}(j))$')}}</th>
+<th>{{KNOWL('ec.conductor_valuation', title='$\mathrm{ord}_p(N)$')}}</th>
+<th>{{KNOWL('ec.discriminant_valuation', title='$\mathrm{ord}_p(\Delta)$')}}</th>
+<th>{{KNOWL('ec.j_invariant_denominator_valuation', title='$\mathrm{ord}_p(\mathrm{den}(j))$')}}</th>
 </tr>
 </thead><tbody>
 {% for pr in data.local_data %}

From 281a580734c6fe5a33300c2a1421c4ba6a654ace Mon Sep 17 00:00:00 2001
From: John Cremona <john.cremona@gmail.com>
Date: Thu, 5 Dec 2024 05:09:49 -0500
Subject: [PATCH 02/11] Hide BMF L-ratio while data is fixed

---
 lmfdb/bianchi_modular_forms/templates/bmf-newform.html | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/lmfdb/bianchi_modular_forms/templates/bmf-newform.html b/lmfdb/bianchi_modular_forms/templates/bmf-newform.html
index 0984fd0ef5..7ff03eae08 100644
--- a/lmfdb/bianchi_modular_forms/templates/bmf-newform.html
+++ b/lmfdb/bianchi_modular_forms/templates/bmf-newform.html
@@ -33,9 +33,9 @@ <h2> {{ KNOWL('mf.bianchi.newform', 'Form') }} </h2>
 <tr><td> {{ KNOWL('mf.bianchi.spaces', title='Newspace')}}:</td><td><a href={{data.newspace_url}}>{{data.newspace_label}}</a> (dimension {{ data.newspace_dimension }}) </td></tr>
 <tr><td> {{ KNOWL('mf.bianchi.sign',title="Sign of functional equation") }}:</td><td> {{data.sign}} </tr>
 <tr><td> {{ KNOWL('mf.bianchi.anr',title="Analytic rank")}}:</td><td> {{data.anrank}}</td> </tr>
-{% if data.Lratio %}
-<tr><td> {{ KNOWL('mf.bianchi.anr',title="L-ratio")}}:</td><td> {{data.Lratio}} </td></tr>
-{% endif %}
+{# See issue #6288
+<tr><td> {{ KNOWL('mf.bianchi.L-ratio',title="L-ratio")}}:</td><td> {{data.Lratio}} </td></tr>
+#}
 </table>
 </p>
 

From d4962b1a27c192a9b00d68b1cffdb6b0b51466c0 Mon Sep 17 00:00:00 2001
From: John Cremona <john.cremona@gmail.com>
Date: Fri, 6 Dec 2024 04:50:36 -0500
Subject: [PATCH 03/11] ECNF local info tweaks and improved discriminant
 display and knowls

---
 lmfdb/ecnf/WebEllipticCurve.py       |  3 ++
 lmfdb/ecnf/templates/ecnf-curve.html | 43 ++++++++++++++++++++++++----
 2 files changed, 40 insertions(+), 6 deletions(-)

diff --git a/lmfdb/ecnf/WebEllipticCurve.py b/lmfdb/ecnf/WebEllipticCurve.py
index 2e86ebb890..bd8651fcc0 100644
--- a/lmfdb/ecnf/WebEllipticCurve.py
+++ b/lmfdb/ecnf/WebEllipticCurve.py
@@ -359,6 +359,9 @@ def make_E(self):
         self.cond_norm = web_latex(self.conductor_norm)
 
         Dnorm = self.normdisc
+        self.model_disc = self.disc.replace('w', Kgen).replace("*","").replace("(","").replace(")","")
+        if Kgen == 'phi':
+            self.model_disc = self.model_disc.replace(Kgen, r"\phi")
         self.disc = pretty_ideal(Kgen, self.disc)
 
         local_data = self.local_data
diff --git a/lmfdb/ecnf/templates/ecnf-curve.html b/lmfdb/ecnf/templates/ecnf-curve.html
index be4782efcf..d5a48fa4d7 100644
--- a/lmfdb/ecnf/templates/ecnf-curve.html
+++ b/lmfdb/ecnf/templates/ecnf-curve.html
@@ -107,7 +107,7 @@ <h2>{{ KNOWL('ec.invariants', title='Invariants')}}</h2>
       <table id="invariants" style="overflow:auto;">
         <tr>
         <td>{{KNOWL('ec.conductor', title='Conductor')}}:</td>
-	<td>$\frak{n}$</td>
+	<td>$\frak{N}$</td>
 	<td>=</td>
         <td>{{ ec.cond }}</td>
         <td>=</td>
@@ -117,7 +117,7 @@ <h2>{{ KNOWL('ec.invariants', title='Invariants')}}</h2>
 
         <tr>
         <td>{{KNOWL('ec.conductor', title='Conductor norm')}}:</td>
-	<td>$N(\frak{n})$</td>
+	<td>$N(\frak{N})$</td>
 	<td>=</td>
         <td>{{ ec.cond_norm }}</td>
         <td>=</td>
@@ -127,7 +127,26 @@ <h2>{{ KNOWL('ec.invariants', title='Invariants')}}</h2>
 
         <tr>
         <td>{{KNOWL('ec.discriminant', title='Discriminant')}}:</td>
-	<td>$(\Delta)$</td>
+	<td>$\Delta$</td>
+	<td>=</td>
+        <td>${{ ec.model_disc }}$</td>
+        </tr>
+
+        <tr>
+          <td>
+	    {% if ec.is_minimal %}
+	    {{KNOWL('ec.minimal_discriminant', title='Discriminant ideal')}}:
+	    {% else %}
+	    {{KNOWL('ec.discriminant', title='Discriminant ideal')}}:
+	    {% endif %}
+	  </td>
+	<td>
+	  {% if ec.is_minimal %}
+	  $\frak{D}_{\mathrm{min}} = (\Delta)$
+	  {% else %}
+	  $(\Delta)$
+	  {% endif %}
+	</td>
 	<td>=</td>
         <td>{{ ec.disc }}</td>
         {% if ec.fact_disc %}
@@ -138,8 +157,20 @@ <h2>{{ KNOWL('ec.invariants', title='Invariants')}}</h2>
 
         <tr><td colspan=4 style="padding:0;">{{ place_code('disc') }}</td></tr>
         <tr>
-        <td>{{KNOWL('ec.discriminant', title='Discriminant norm')}}:</td>
-	<td>$N(\Delta)$</td>
+          <td>
+	    {% if ec.is_minimal %}
+	    {{KNOWL('ec.minimal_discriminant', title='Discriminant norm')}}:
+	    {% else %}
+	    {{KNOWL('ec.discriminant', title='Discriminant norm')}}:
+            {% endif %}
+	  </td>
+	<td>
+	  {% if ec.is_minimal %}
+	  $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$
+	  {% else %}
+	  $N(\Delta)$
+	  {% endif %}
+	</td>
 	<td>=</td>
         <td>{{ ec.disc_norm }}</td>
         <td>=</td>
@@ -363,7 +394,7 @@ <h2>{{KNOWL('ec.local_data', title='Local data')}} at {{KNOWL('ec.bad_reduction'
 
 
 {% if ec.local_data %}
-<table class="ntdata"><thead>
+<table class="ntdata centered"><thead>
 <tr>
 <th>$\mathfrak{p}$</th>
 <th>$N(\mathfrak{p})$</th>

From 3b031bbac9e8fba0b27dc828ceb4ff3823d16dd8 Mon Sep 17 00:00:00 2001
From: John Cremona <john.cremona@gmail.com>
Date: Thu, 5 Dec 2024 05:09:49 -0500
Subject: [PATCH 04/11] Hide BMF L-ratio while data is fixed

---
 lmfdb/bianchi_modular_forms/templates/bmf-newform.html | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/lmfdb/bianchi_modular_forms/templates/bmf-newform.html b/lmfdb/bianchi_modular_forms/templates/bmf-newform.html
index 0984fd0ef5..7ff03eae08 100644
--- a/lmfdb/bianchi_modular_forms/templates/bmf-newform.html
+++ b/lmfdb/bianchi_modular_forms/templates/bmf-newform.html
@@ -33,9 +33,9 @@ <h2> {{ KNOWL('mf.bianchi.newform', 'Form') }} </h2>
 <tr><td> {{ KNOWL('mf.bianchi.spaces', title='Newspace')}}:</td><td><a href={{data.newspace_url}}>{{data.newspace_label}}</a> (dimension {{ data.newspace_dimension }}) </td></tr>
 <tr><td> {{ KNOWL('mf.bianchi.sign',title="Sign of functional equation") }}:</td><td> {{data.sign}} </tr>
 <tr><td> {{ KNOWL('mf.bianchi.anr',title="Analytic rank")}}:</td><td> {{data.anrank}}</td> </tr>
-{% if data.Lratio %}
-<tr><td> {{ KNOWL('mf.bianchi.anr',title="L-ratio")}}:</td><td> {{data.Lratio}} </td></tr>
-{% endif %}
+{# See issue #6288
+<tr><td> {{ KNOWL('mf.bianchi.L-ratio',title="L-ratio")}}:</td><td> {{data.Lratio}} </td></tr>
+#}
 </table>
 </p>
 

From 033a961575b226ebb1aadde4ef5e90ff136ac741 Mon Sep 17 00:00:00 2001
From: John Cremona <john.cremona@gmail.com>
Date: Thu, 5 Dec 2024 09:57:31 -0500
Subject: [PATCH 05/11] ECNF: added Neron-Tate regulator to curve home page

---
 lmfdb/ecnf/WebEllipticCurve.py       | 13 +++++++++++--
 lmfdb/ecnf/templates/ecnf-curve.html | 20 +++++++++++++++++++-
 2 files changed, 30 insertions(+), 3 deletions(-)

diff --git a/lmfdb/ecnf/WebEllipticCurve.py b/lmfdb/ecnf/WebEllipticCurve.py
index bd8651fcc0..ea5f8c65df 100644
--- a/lmfdb/ecnf/WebEllipticCurve.py
+++ b/lmfdb/ecnf/WebEllipticCurve.py
@@ -566,15 +566,24 @@ def make_E(self):
         if self.bsd_status in ["conditional", "unconditional"]:
             if self.ar == 0:
                 self.reg = web_latex(1)  # otherwise we only get 1.00000...
+                self.NTreg = web_latex(1)  # otherwise we only get 1.00000...
             else:
                 try:
-                    self.reg = web_latex(self.reg)
+                    R = self.reg
+                    self.reg = web_latex(R)
+                    self.NTreg = web_latex(R * K.degree()**self.rank)
                 except AttributeError:
                     self.reg = "not available"
+                    self.NTreg = "not available"
         elif self.rk != "not available":
-            self.reg = web_latex(self.reg) if self.rank else web_latex(1)
+            R = self.reg
+            self.reg = web_latex(R) if self.rank else web_latex(1)
+            self.NTreg = web_latex(R * K.degree()**self.rank) if self.rank else web_latex(1)
         else:
             self.reg = "not available"
+            self.NTreg = "not available"
+        print(f"reg = {self.reg}")
+        print(f"NTreg = {self.NTreg}")
 
         # Generators
         try:
diff --git a/lmfdb/ecnf/templates/ecnf-curve.html b/lmfdb/ecnf/templates/ecnf-curve.html
index d5a48fa4d7..6d0a9ab19b 100644
--- a/lmfdb/ecnf/templates/ecnf-curve.html
+++ b/lmfdb/ecnf/templates/ecnf-curve.html
@@ -296,7 +296,7 @@ <h2> {{ KNOWL('ec.bsdconjecture', title='BSD invariants') }}</h2>
 	    {% endif %}
 	  </td>
 	  <td>$\mathrm{Reg}(E/K)$</td>
-	  <td>{% if ec.rk == 0 %}={% else %}&approx;{% endif %}</td>
+	  <td>{% if ec.rank == 0 %}={% else %}&approx;{% endif %}</td>
           <td>
 	  {% if ec.reg=='not available' %}
 	  not available
@@ -305,6 +305,24 @@ <h2> {{ KNOWL('ec.bsdconjecture', title='BSD invariants') }}</h2>
 	  {% endif %}
 	</td>
         </tr>
+        <tr>
+	  <td align='left'>
+	    {% if ec.bsd_status == "conditional" %}
+            {{ KNOWL('ec.regulator', title='N&eacute;ron-Tate Regulator*') }}:
+	    {% else %}
+	    {{ KNOWL('ec.regulator', title='N&eacute;ron-Tate Regulator') }}:
+	    {% endif %}
+	  </td>
+	  <td>$\mathrm{Reg}_{\mathrm{NT}}(E/K)$</td>
+	  <td>{% if ec.rank == 0 %}={% else %}&approx;{% endif %}</td>
+          <td>
+	  {% if ec.NTreg=='not available' %}
+	  not available
+	  {% else %}
+	  {{ ec.NTreg }}
+	  {% endif %}
+	</td>
+        </tr>
 
         <tr>
         <td align='left'>{{ KNOWL('ec.period', title='Global period') }}:</td>

From 75f8cbc373696ae29a74c02894e51a0ba332435f Mon Sep 17 00:00:00 2001
From: John Cremona <john.cremona@gmail.com>
Date: Thu, 5 Dec 2024 12:17:40 -0500
Subject: [PATCH 06/11] ECNF: added BSD formula to curve home page where
 possible

---
 lmfdb/ecnf/WebEllipticCurve.py       | 73 +++++++++++++++++++++++-----
 lmfdb/ecnf/templates/ecnf-curve.html | 23 ++++++++-
 2 files changed, 81 insertions(+), 15 deletions(-)

diff --git a/lmfdb/ecnf/WebEllipticCurve.py b/lmfdb/ecnf/WebEllipticCurve.py
index ea5f8c65df..2c6838e3cb 100644
--- a/lmfdb/ecnf/WebEllipticCurve.py
+++ b/lmfdb/ecnf/WebEllipticCurve.py
@@ -2,7 +2,7 @@
 from flask import url_for
 from urllib.parse import quote
 from markupsafe import Markup, escape
-from sage.all import (Infinity, PolynomialRing, QQ, RDF, ZZ, KodairaSymbol,
+from sage.all import (Infinity, PolynomialRing, QQ, RDF, ZZ, RR, KodairaSymbol,
                       implicit_plot, plot, prod, rainbow, sqrt, text, var)
 from lmfdb import db
 from lmfdb.utils import (encode_plot, names_and_urls, web_latex, display_knowl,
@@ -494,7 +494,8 @@ def make_E(self):
             self.mw_struct = "unknown"
 
         # Torsion
-        self.ntors = web_latex(self.torsion_order)
+        BSDntors = self.torsion_order
+        self.ntors = web_latex(BSDntors)
         self.tr = len(self.torsion_structure)
         if self.tr == 0:
             self.tor_struct_pretty = "$0$"
@@ -563,27 +564,28 @@ def make_E(self):
                 self.bsd_status = "missing_gens"
 
         # Regulator only in conditional/unconditional cases, or when we know the rank:
+        BSDReg = None
         if self.bsd_status in ["conditional", "unconditional"]:
             if self.ar == 0:
-                self.reg = web_latex(1)  # otherwise we only get 1.00000...
-                self.NTreg = web_latex(1)  # otherwise we only get 1.00000...
+                BSDReg = 1
+                self.reg = self.NTreg = web_latex(BSDReg)  # otherwise we only get 1.00000...
             else:
                 try:
                     R = self.reg
+                    BSDReg = R * K.degree()**self.rank
                     self.reg = web_latex(R)
-                    self.NTreg = web_latex(R * K.degree()**self.rank)
+                    self.NTreg = web_latex(BSDReg)
                 except AttributeError:
                     self.reg = "not available"
                     self.NTreg = "not available"
         elif self.rk != "not available":
             R = self.reg
+            BSDReg = R * K.degree()**self.rank
             self.reg = web_latex(R) if self.rank else web_latex(1)
-            self.NTreg = web_latex(R * K.degree()**self.rank) if self.rank else web_latex(1)
+            self.NTreg = web_latex(BSDReg) if self.rank else web_latex(1)
         else:
             self.reg = "not available"
             self.NTreg = "not available"
-        print(f"reg = {self.reg}")
-        print(f"NTreg = {self.NTreg}")
 
         # Generators
         try:
@@ -594,17 +596,24 @@ def make_E(self):
             self.gens = []
             self.gens_and_heights = []
 
-        # Global period
+        # Global period -- see issue #5409 for why we multiply by 2**nc
+        BSDomega = None
         try:
-            self.omega = web_latex(self.omega)
+            BSDomega = self.omega
+            nc = self.signature[1] # number of complex places
+            if nc:
+                BSDomega *= 2**nc
+            self.omega = web_latex(BSDomega)
         except AttributeError:
             self.omega = "not available"
 
         # L-value
+        BSDLvalue = None
         try:
             r = int(self.analytic_rank)
             # lhs = "L(E,1) = " if r==0 else "L'(E,1) = " if r==1 else "L^{{({})}}(E,1)/{}! = ".format(r,r)
-            self.Lvalue = web_latex(self.Lvalue)
+            BSDLvalue = self.Lvalue
+            self.Lvalue = web_latex(BSDLvalue)
         except (TypeError, AttributeError):
             self.Lvalue = "not available"
 
@@ -616,14 +625,52 @@ def make_E(self):
             self.tamagawa_factors = r'\cdot'.join(cp_fac)
         else:
             self.tamagawa_factors = None
-        self.tamagawa_product = web_latex(prod(tamagawa_numbers,1))
+        BSDprodcp = prod(tamagawa_numbers,1)
+        self.tamagawa_product = web_latex(BSDprodcp)
 
         # Analytic Sha
+        BSDsha = None
         try:
-            self.sha = web_latex(self.sha) + " (rounded)"
+            BSDsha = self.sha
+            self.sha = web_latex(BSDsha) + " (rounded)"
         except AttributeError:
             self.sha = "not available"
 
+        # Check analytic Sha value compatible with formula in the knowl (see issue #5409)
+
+        BSDrootdisc = RR(K.discriminant().abs()).sqrt()
+        if BSDLvalue and BSDsha and BSDReg and BSDsha:
+            BSDsha_numerator = BSDrootdisc * BSDntors**2
+            BSDsha_denominator = BSDReg * BSDomega * BSDprodcp
+            BSDsha_from_formula = BSDLvalue * BSDsha_numerator / BSDsha_denominator
+            BSDLvalue_from_formula = BSDsha * BSDsha_denominator / BSDsha_numerator
+            self.BSDsha = web_latex(BSDsha_from_formula)
+            self.BSDLvalue = web_latex(BSDLvalue_from_formula)
+        else:
+            self.BSDsha = "not available"
+            self.BSDLvalue = "not available"
+
+        # The BSD formula for display
+
+        dot = '\\cdot'
+        approx = '\\approx'
+        frac = '\\frac'
+        Sha = '\\# &#1064;(E/K)'
+        Om = '\\Omega(E/K)'
+        Reg = '\\mathrm{Reg}_{\\mathrm{NT}}(E/K)'
+        prodcp = '\\prod_{\\mathfrak{p}} c_{\\mathfrak{p}}'
+        tors2 = '\\#E(K)_{\\mathrm{tor}}^2'
+        rootD = '\\left|D_K\\right|'
+
+        lder_name  = rf"L^{({r})}(E,1)/{r}!" if r>=2 else "L'(E,1)" if r else "L(E,1)"
+        lhs_num    = rf'{Sha} {dot} {Om} {dot} {Reg} {dot} {prodcp}'
+        lhs_den    = rf'{tors2} {dot} {rootD}'
+        lhs        = rf'{frac}{{ {lhs_num} }} {{ {lhs_den} }}'
+        rhs_num    = rf'{BSDsha} {dot} {BSDomega:0.6f} {dot} {BSDReg:0.6f} {dot} {BSDprodcp}'
+        rhs_den    = rf'{{{BSDntors}^2 {dot} {BSDrootdisc:0.6f}}}'
+        rhs        = rf'{frac}{{ {rhs_num} }} {{ {rhs_den} }}'
+        self.bsd_formula = rf'{BSDLvalue:0.9f} {approx} {lder_name} = {lhs} {approx} {rhs} {approx} {BSDLvalue_from_formula:0.9f}'
+
         # Local data
 
         # The Kodaira symbol is stored as an int in pari encoding. The
diff --git a/lmfdb/ecnf/templates/ecnf-curve.html b/lmfdb/ecnf/templates/ecnf-curve.html
index 6d0a9ab19b..03442abe92 100644
--- a/lmfdb/ecnf/templates/ecnf-curve.html
+++ b/lmfdb/ecnf/templates/ecnf-curve.html
@@ -367,8 +367,19 @@ <h2> {{ KNOWL('ec.bsdconjecture', title='BSD invariants') }}</h2>
 	  <td>&#1064;${}_{\mathrm{an}}$</td><td>=</td>
           <td> {{ ec.sha }} </td>
         </tr>
+{#
+	<tr>
+	  <td align='left'>
+	    {{ KNOWL('ec.analytic_sha_order', title='Analytic order of &#1064; (from formula)') }}:
+	  </td>
+	  <td>&#1064;${}_{\mathrm{an}}$</td><td>=</td>
+          <td> {{ ec.BSDsha }} </td>
+        </tr>
+#}
         </table>
-    </p>
+{# %%%%%%%%%%%%%%%% END OF TABLE %%%%%%%%%%%%%%%%%%% #}
+</p>
+
 {% if ec.bsd_status == "conditional" %}
 <p>
   * Conditional on {{ KNOWL('ec.bsdconjecture', title='BSD') }}: assuming rank = analytic rank.
@@ -382,8 +393,16 @@ <h2> {{ KNOWL('ec.bsdconjecture', title='BSD invariants') }}</h2>
 </p>
 {% endif %}
 {% endif %}
+</div>
 
-{# %%%%%%%%%%%%%%%% END OF TABLE %%%%%%%%%%%%%%%%%%% #}
+<h2> {{ KNOWL('ec.bsdconjecture', title='BSD formula') }}</h2>
+<div>
+<p style="margin:10px 280px;">
+$\displaystyle {{ ec.bsd_formula|safe }}$
+</p>
+<center>
+{{ place_code('bsd_formula') }}
+</center>
 </div>
 
 <h2>{{KNOWL('ec.local_data', title='Local data')}} at {{KNOWL('ec.bad_reduction', title='primes of bad reduction')}} </h2>

From 96e17b9d87ad07ebcc6ef8de86aed909dfff8833 Mon Sep 17 00:00:00 2001
From: John Cremona <john.cremona@gmail.com>
Date: Fri, 6 Dec 2024 05:21:23 -0500
Subject: [PATCH 07/11] ECNF: BSD formula tweaks, no display or error when not
 all data available

---
 lmfdb/ecnf/WebEllipticCurve.py       | 49 +++++++++++++++-------------
 lmfdb/ecnf/templates/ecnf-curve.html |  6 +++-
 2 files changed, 32 insertions(+), 23 deletions(-)

diff --git a/lmfdb/ecnf/WebEllipticCurve.py b/lmfdb/ecnf/WebEllipticCurve.py
index 2c6838e3cb..8ce2afac53 100644
--- a/lmfdb/ecnf/WebEllipticCurve.py
+++ b/lmfdb/ecnf/WebEllipticCurve.py
@@ -639,38 +639,43 @@ def make_E(self):
         # Check analytic Sha value compatible with formula in the knowl (see issue #5409)
 
         BSDrootdisc = RR(K.discriminant().abs()).sqrt()
-        if BSDLvalue and BSDsha and BSDReg and BSDsha:
+        if BSDLvalue and BSDsha and BSDReg and (self.rank is not None):
             BSDsha_numerator = BSDrootdisc * BSDntors**2
             BSDsha_denominator = BSDReg * BSDomega * BSDprodcp
             BSDsha_from_formula = BSDLvalue * BSDsha_numerator / BSDsha_denominator
             BSDLvalue_from_formula = BSDsha * BSDsha_denominator / BSDsha_numerator
             self.BSDsha = web_latex(BSDsha_from_formula)
             self.BSDLvalue = web_latex(BSDLvalue_from_formula)
+
+            # The BSD formula for display
+
+            dot = '\\cdot'
+            approx = '\\approx'
+            frac = '\\frac'
+            Sha = '\\# &#1064;(E/K)'
+            Om = '\\Omega(E/K)'
+            Reg = '\\mathrm{Reg}_{\\mathrm{NT}}(E/K)'
+            prodcp = '\\prod_{\\mathfrak{p}} c_{\\mathfrak{p}}'
+            tors2 = '\\#E(K)_{\\mathrm{tor}}^2'
+            rootD = '\\left|d_K\\right|^{1/2}'
+
+            lder_name  = rf"L^{{({r})}}(E/K,1)/{r}!" if r>=2 else "L'(E/K,1)" if r else "L(E/K,1)"
+            lhs_num    = rf'{Sha} {dot} {Om} {dot} {Reg} {dot} {prodcp}'
+            lhs_den    = rf'{tors2} {dot} {rootD}'
+            lhs        = rf'{frac}{{ {lhs_num} }} {{ {lhs_den} }}'
+            rhs_num    = rf'{BSDsha} {dot} {BSDomega:0.6f} {dot} {BSDReg} {dot} {BSDprodcp}'
+            if r:
+                rhs_num    = rf'{BSDsha} {dot} {BSDomega:0.6f} {dot} {BSDReg:0.6f} {dot} {BSDprodcp}'
+            rhs_den    = rf'{{{BSDntors}^2 {dot} {BSDrootdisc:0.6f}}}'
+            rhs        = rf'{frac}{{ {rhs_num} }} {{ {rhs_den} }}'
+            self.bsd_formula = rf'{BSDLvalue:0.9f} {approx} {lder_name} = {lhs} {approx} {rhs} {approx} {BSDLvalue_from_formula:0.9f}'
+
         else:
             self.BSDsha = "not available"
             self.BSDLvalue = "not available"
+            self.bsd_formula = None
 
-        # The BSD formula for display
-
-        dot = '\\cdot'
-        approx = '\\approx'
-        frac = '\\frac'
-        Sha = '\\# &#1064;(E/K)'
-        Om = '\\Omega(E/K)'
-        Reg = '\\mathrm{Reg}_{\\mathrm{NT}}(E/K)'
-        prodcp = '\\prod_{\\mathfrak{p}} c_{\\mathfrak{p}}'
-        tors2 = '\\#E(K)_{\\mathrm{tor}}^2'
-        rootD = '\\left|D_K\\right|'
-
-        lder_name  = rf"L^{({r})}(E,1)/{r}!" if r>=2 else "L'(E,1)" if r else "L(E,1)"
-        lhs_num    = rf'{Sha} {dot} {Om} {dot} {Reg} {dot} {prodcp}'
-        lhs_den    = rf'{tors2} {dot} {rootD}'
-        lhs        = rf'{frac}{{ {lhs_num} }} {{ {lhs_den} }}'
-        rhs_num    = rf'{BSDsha} {dot} {BSDomega:0.6f} {dot} {BSDReg:0.6f} {dot} {BSDprodcp}'
-        rhs_den    = rf'{{{BSDntors}^2 {dot} {BSDrootdisc:0.6f}}}'
-        rhs        = rf'{frac}{{ {rhs_num} }} {{ {rhs_den} }}'
-        self.bsd_formula = rf'{BSDLvalue:0.9f} {approx} {lder_name} = {lhs} {approx} {rhs} {approx} {BSDLvalue_from_formula:0.9f}'
-
+        print(f"BSD: {self.bsd_formula}")
         # Local data
 
         # The Kodaira symbol is stored as an int in pari encoding. The
diff --git a/lmfdb/ecnf/templates/ecnf-curve.html b/lmfdb/ecnf/templates/ecnf-curve.html
index 03442abe92..e288520570 100644
--- a/lmfdb/ecnf/templates/ecnf-curve.html
+++ b/lmfdb/ecnf/templates/ecnf-curve.html
@@ -274,7 +274,7 @@ <h2> {{ KNOWL('ec.bsdconjecture', title='BSD invariants') }}</h2>
 	<tr>
 	  <td align='left'>{{ KNOWL('ec.rank', title="Mordell-Weil rank")}}:</td>
 	  {% if ec.rk == "not available" %}
-	  <td>$r$?</td>
+	  <td>$r?$</td>
 	  <td>&nbsp;</td>
 	  {% if ec.rank_bounds != "not available" %}
 	  <td>\({{ ec.rk_lb }} \le r \le {{ec.rk_ub}}\)</td>
@@ -395,6 +395,8 @@ <h2> {{ KNOWL('ec.bsdconjecture', title='BSD invariants') }}</h2>
 {% endif %}
 </div>
 
+{% if ec.bsd_formula %}
+
 <h2> {{ KNOWL('ec.bsdconjecture', title='BSD formula') }}</h2>
 <div>
 <p style="margin:10px 280px;">
@@ -405,6 +407,8 @@ <h2> {{ KNOWL('ec.bsdconjecture', title='BSD formula') }}</h2>
 </center>
 </div>
 
+{% endif %}
+
 <h2>{{KNOWL('ec.local_data', title='Local data')}} at {{KNOWL('ec.bad_reduction', title='primes of bad reduction')}} </h2>
 <div>
  {{ place_code('localdata') }}

From 2ccd9ec9e2ed315329417e77997edea8a4f08d64 Mon Sep 17 00:00:00 2001
From: John Cremona <john.cremona@gmail.com>
Date: Fri, 6 Dec 2024 06:38:48 -0500
Subject: [PATCH 08/11] ECNF: adjust global period where necessary

---
 lmfdb/ecnf/WebEllipticCurve.py | 26 ++++++++++++++++++++++++--
 1 file changed, 24 insertions(+), 2 deletions(-)

diff --git a/lmfdb/ecnf/WebEllipticCurve.py b/lmfdb/ecnf/WebEllipticCurve.py
index 8ce2afac53..47dbb5c17d 100644
--- a/lmfdb/ecnf/WebEllipticCurve.py
+++ b/lmfdb/ecnf/WebEllipticCurve.py
@@ -596,7 +596,14 @@ def make_E(self):
             self.gens = []
             self.gens_and_heights = []
 
-        # Global period -- see issue #5409 for why we multiply by 2**nc
+        # Global period -- see issue #5409 for why we multiply by
+        # 2**nc in most cases.  However, data computed after
+        # 2024-07-09 (including all data for imaginary quadratic
+        # fields of absolute discriminant > 600 as well as some larger
+        # conductors for other IQFs) already has the extra factor of
+        # 2.  As a fail-safe until we fix the data in all cases, we
+        # will test (using the BSD formula) whether to remove the
+        # factor of 2 added here.
         BSDomega = None
         try:
             BSDomega = self.omega
@@ -639,10 +646,23 @@ def make_E(self):
         # Check analytic Sha value compatible with formula in the knowl (see issue #5409)
 
         BSDrootdisc = RR(K.discriminant().abs()).sqrt()
+        BSDok = True
         if BSDLvalue and BSDsha and BSDReg and (self.rank is not None):
             BSDsha_numerator = BSDrootdisc * BSDntors**2
             BSDsha_denominator = BSDReg * BSDomega * BSDprodcp
             BSDsha_from_formula = BSDLvalue * BSDsha_numerator / BSDsha_denominator
+            BSDok = BSDsha_from_formula.round() == BSDsha
+            if not BSDok:
+                # this means that we doubled BSDomega when we should
+                # not have, so BSDsha_denominator is doubled and
+                # BSDsha_from formula is halved
+                print(f"BSD normalization: adjusting Omega for {self.label}: stored Sha = {BSDsha} but formula gives {BSDsha_from_formula} which rounds to {BSDsha_from_formula.round()}")
+                BSDok = ((BSDsha/BSDsha_from_formula)-2).abs() < 0.01
+                if not BSDok:
+                    print(f"BSD normalization issue with {self.label}: stored Sha = {BSDsha} but formula gives {BSDsha_from_formula}")
+                BSDomega /= 2
+                BSDsha_denominator /= 2
+                BSDsha_from_formula *= 2
             BSDLvalue_from_formula = BSDsha * BSDsha_denominator / BSDsha_numerator
             self.BSDsha = web_latex(BSDsha_from_formula)
             self.BSDLvalue = web_latex(BSDLvalue_from_formula)
@@ -675,7 +695,9 @@ def make_E(self):
             self.BSDLvalue = "not available"
             self.bsd_formula = None
 
-        print(f"BSD: {self.bsd_formula}")
+        if not BSDok: # don't display the formula if it is not correct
+            self.bsd_formula = None
+
         # Local data
 
         # The Kodaira symbol is stored as an int in pari encoding. The

From 3a28ab127aa1ac2061412f1d97e0df2133592604 Mon Sep 17 00:00:00 2001
From: Edgar Costa <edgarc@mit.edu>
Date: Mon, 9 Dec 2024 14:51:21 +1300
Subject: [PATCH 09/11] no need for redirect, as we are expected to return a
 url

---
 lmfdb/abvar/fq/main.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/lmfdb/abvar/fq/main.py b/lmfdb/abvar/fq/main.py
index ce8076c294..1f869f5c4a 100644
--- a/lmfdb/abvar/fq/main.py
+++ b/lmfdb/abvar/fq/main.py
@@ -146,7 +146,7 @@ def url_for_label(label):
         validate_label(label)
     except ValueError as err:
         flash_error("%s is not a valid label: %s.", label, str(err))
-        return redirect(url_for(".abelian_varieties"))
+        return url_for(".abelian_varieties")
     g, q, iso = split_label(label)
     return url_for(".abelian_varieties_by_gqi", g=g, q=q, iso=iso)
 

From ec8312b89419fbe8c1c73b55d4ca6be71752b6e8 Mon Sep 17 00:00:00 2001
From: John Cremona <john.cremona@gmail.com>
Date: Mon, 9 Dec 2024 06:03:40 -0500
Subject: [PATCH 10/11] add ? to = in BSD formula

---
 lmfdb/ecnf/WebEllipticCurve.py | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/lmfdb/ecnf/WebEllipticCurve.py b/lmfdb/ecnf/WebEllipticCurve.py
index 47dbb5c17d..7b3ab91cba 100644
--- a/lmfdb/ecnf/WebEllipticCurve.py
+++ b/lmfdb/ecnf/WebEllipticCurve.py
@@ -671,6 +671,7 @@ def make_E(self):
 
             dot = '\\cdot'
             approx = '\\approx'
+            eq_query = '\\overset{?}{=}'
             frac = '\\frac'
             Sha = '\\# &#1064;(E/K)'
             Om = '\\Omega(E/K)'
@@ -688,7 +689,7 @@ def make_E(self):
                 rhs_num    = rf'{BSDsha} {dot} {BSDomega:0.6f} {dot} {BSDReg:0.6f} {dot} {BSDprodcp}'
             rhs_den    = rf'{{{BSDntors}^2 {dot} {BSDrootdisc:0.6f}}}'
             rhs        = rf'{frac}{{ {rhs_num} }} {{ {rhs_den} }}'
-            self.bsd_formula = rf'{BSDLvalue:0.9f} {approx} {lder_name} = {lhs} {approx} {rhs} {approx} {BSDLvalue_from_formula:0.9f}'
+            self.bsd_formula = rf'{BSDLvalue:0.9f} {approx} {lder_name} {eq_query} {lhs} {approx} {rhs} {approx} {BSDLvalue_from_formula:0.9f}'
 
         else:
             self.BSDsha = "not available"

From 81ac07e1d253857547e5caf25011996b19e5a4eb Mon Sep 17 00:00:00 2001
From: John Cremona <john.cremona@gmail.com>
Date: Mon, 9 Dec 2024 09:34:57 -0500
Subject: [PATCH 11/11] ECNF: fixing the global period adjustment fix

---
 lmfdb/ecnf/WebEllipticCurve.py | 10 ++++++++--
 1 file changed, 8 insertions(+), 2 deletions(-)

diff --git a/lmfdb/ecnf/WebEllipticCurve.py b/lmfdb/ecnf/WebEllipticCurve.py
index 7b3ab91cba..22b64cb83b 100644
--- a/lmfdb/ecnf/WebEllipticCurve.py
+++ b/lmfdb/ecnf/WebEllipticCurve.py
@@ -651,18 +651,24 @@ def make_E(self):
             BSDsha_numerator = BSDrootdisc * BSDntors**2
             BSDsha_denominator = BSDReg * BSDomega * BSDprodcp
             BSDsha_from_formula = BSDLvalue * BSDsha_numerator / BSDsha_denominator
-            BSDok = BSDsha_from_formula.round() == BSDsha
+            BSDsha_from_formula_rounded = BSDsha_from_formula.round()
+            BSDok = (BSDsha_from_formula_rounded == BSDsha) and ((BSDsha_from_formula_rounded -BSDsha_from_formula).abs() < 0.001)
+            #print(f"{BSDsha_from_formula=}")
+            #print(f"{BSDsha_from_formula_rounded=}")
+            #print(f"{BSDsha=}")
+            #print(f"{BSDok=}")
             if not BSDok:
                 # this means that we doubled BSDomega when we should
                 # not have, so BSDsha_denominator is doubled and
                 # BSDsha_from formula is halved
-                print(f"BSD normalization: adjusting Omega for {self.label}: stored Sha = {BSDsha} but formula gives {BSDsha_from_formula} which rounds to {BSDsha_from_formula.round()}")
+                print(f"BSD normalization: adjusting Omega for {self.label}: stored Sha = {BSDsha} but formula gives {BSDsha_from_formula}")
                 BSDok = ((BSDsha/BSDsha_from_formula)-2).abs() < 0.01
                 if not BSDok:
                     print(f"BSD normalization issue with {self.label}: stored Sha = {BSDsha} but formula gives {BSDsha_from_formula}")
                 BSDomega /= 2
                 BSDsha_denominator /= 2
                 BSDsha_from_formula *= 2
+                BSDsha_from_formula_rounded = BSDsha_from_formula.round()
             BSDLvalue_from_formula = BSDsha * BSDsha_denominator / BSDsha_numerator
             self.BSDsha = web_latex(BSDsha_from_formula)
             self.BSDLvalue = web_latex(BSDLvalue_from_formula)