Merge pull request #6931 from rvisser7/dirichlet-patch

Small improvements to Dirichlet characters

Author: roed314

lmfdb/characters/code.yaml

@@ -94,6 +94,16 @@ galois_orbit:
     order := Order(chi);
     { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 
+value_field:
+  comment: Field of values of chi
+  sage: CyclotomicField(chi.multiplicative_order())
+  pari: nfinit(polcyclo(charorder(g,chi)))
+  magma: CyclotomicField(Order(chi));
+
+kernel_field:
+  comment: Fixed field
+  sage: chi.fixed_field()
+
 gauss_sum:
   comment: Gauss sum
   sage: chi.gauss_sum(a)
@@ -109,8 +119,9 @@ kloosterman_sum:
 
 value:
   comment: Value of chi at x
-  sage: chi(x) # x integer
-  pari: chareval(g,chi,x) \\\\ x integer, value in Q/Z'
+  sage: |
+    chi(x)  # x integer
+  pari: chareval(g,chi,x)  \\ x integer, value in Q/Z
   magma: chi(x)
 
 

lmfdb/characters/main.py

@@ -308,7 +308,6 @@ def dirichlet_character_search(info, query):
     common_parse(info, query)
 
 
-@characters_page.route("/Dirichlet")
 @characters_page.route("/Dirichlet/")
 def render_DirichletNavigation():
     if request.args:
@@ -416,7 +415,6 @@ def make_webchar(args, get_bread=False):
         return WebSmallDirichletCharacter(**args)
 
 
-@characters_page.route("/Dirichlet/<modulus>")
 @characters_page.route("/Dirichlet/<modulus>/")
 @characters_page.route("/Dirichlet/<int:modulus>/<int:number>")
 @characters_page.route("/Dirichlet/<int:modulus>/<orbit_label>")  # orbit_label is a Cremona_letter_code identifying the orbit
@@ -557,6 +555,8 @@ def dirchar_code_download(label, download_type):
     Render a text page to download all Magma/Sage/PariGP code snippets for the various Dirichlet character pages
     Returns code snippets for either the individual Dirichet character, character orbit, or character group, depending on label
     """
+    sorted_code_names = ['character_init', 'kronecker_symbol', 'modulus', 'conductor', 'order',
+                         'is_real', 'is_primitive', 'parity', 'value_field', 'kernel_field']
     try:
         if label.count(".") == 0:
             # Group of Dirichlet characters
@@ -567,12 +567,11 @@ def dirchar_code_download(label, download_type):
             # Orbit of Dirichlet characters
             modulus, orbit_label = label.split(".")
             dc = make_webchar({'type':'Dirichlet', 'modulus':modulus, 'orbit_label':orbit_label})
-            sorted_code_names = ['character_init', 'kronecker_symbol', 'modulus', 'conductor', 'order', 'is_real', 'is_primitive', 'parity']
         elif label.count(".") == 2:
             # Individual Dirichlet character
             modulus, orbit_label, number = label.split(".")
             dc = make_webchar({'type':'Dirichlet', 'modulus':modulus, 'orbit_label':orbit_label, 'number':number})
-            sorted_code_names = ['character_init', 'kronecker_symbol', 'modulus', 'conductor', 'order', 'is_real', 'is_primitive', 'parity', 'galois_orbit']
+            sorted_code_names.insert(8, 'galois_orbit')
         else:
             return abort(404, f"Invalid label {label}")
         if label.count(".") > 0 and dc.symbol_numerator() is None:

lmfdb/characters/templates/CharacterCommon.html

@@ -83,14 +83,13 @@ <h2>
   {% if valuefield %}
   <tr>
     <td> {{KNOWL('character.dirichlet.value_field',title='Field of values')}}:</td>
-    <td> {{ valuefield | safe }} </td>
+    <td> {{ valuefield | safe }} </td> <td>{{ place_code('value_field') }}</td>
   </tr>
   {% endif %}
   {% if kerfield %}
   <tr>
     <td>{{ KNOWL('character.dirichlet.field_cut_out',title='Fixed field') }}:</td>
-    <td> {{ kerfield | safe }}
-    </td>
+    <td> {{ kerfield | safe }} </td> <td>{{ place_code('kernel_field') }}</td>
   </tr>
   {% endif %}
 </table>

lmfdb/characters/web_character.py

@@ -800,6 +800,13 @@ def title(self):
     def order(self):
         return euler_phi(self.modulus)
 
+    @cached_method
+    def code_snippets(self):
+        code = super().code_snippets()
+        # Adjust frontmatter for constructing group of Dirichlet characters
+        code["frontmatter"]["all"] = code["frontmatter"]["all"].replace("character", "character group of modulus")
+        return code
+
 
 class WebDBDirichletGroup(WebDirichletGroup, WebDBDirichlet):
     """
@@ -994,6 +1001,8 @@ def code_snippets(self):
         # Sage code in special case for modulus 1
         if self.modulus == 1:
             self._genvalues_for_code = []
+            # Sage throws error for "chi.fixed_field()" if modulus is 1
+            code['kernel_field'].pop('sage')
 
         data = {'modulus': self.modulus, 'number' : self.number,
                 'symbol_num' : self.symbol_numerator(),

lmfdb/tests/snippet_tests/characters/code-1.a.1-gp.log

@@ -17,4 +17,6 @@ gp> order = charorder(g,chi)
 1
 gp> [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 []
+gp> nfinit(polcyclo(charorder(g,chi)))
+[x - 1, [1, 0], 1, 1, [Mat(1), Mat(1), Mat(16), Mat(1), 1, Mat(1), [1, 0], []], [1.0000000000000000000000000000000000000], [1], Mat(1), Mat(1)]
 gp> 

lmfdb/tests/snippet_tests/characters/code-1.a.1-sage.log

@@ -17,4 +17,6 @@ sage: chi.is_odd()
 False
 sage: chi.galois_orbit()
 [Dirichlet character modulo 1 of conductor 1]
+sage: CyclotomicField(chi.multiplicative_order())
+Cyclotomic Field of order 1 and degree 1
 sage: 

lmfdb/tests/snippet_tests/characters/code-3.b-gp.log

@@ -19,4 +19,6 @@ gp> #znconreyconductor(g,chi)==1
 1
 gp> zncharisodd(g,chi)
 1
+gp> nfinit(polcyclo(charorder(g,chi)))
+[x + 1, [1, 0], 1, 1, [Mat(1), Mat(1), Mat(16), Mat(1), 1, Mat(1), [1, 0], []], [-1.0000000000000000000000000000000000000], [1], Mat(1), Mat(1)]
 gp> 

lmfdb/tests/snippet_tests/characters/code-3.b-sage.log

@@ -19,4 +19,8 @@ sage: chi.is_primitive()
 True
 sage: chi.is_odd()
 True
+sage: CyclotomicField(chi.multiplicative_order())
+Cyclotomic Field of order 2 and degree 1
+sage: chi.fixed_field()
+Number Field in a with defining polynomial x^2 + x + 1
 sage: 

lmfdb/tests/snippet_tests/characters/code-5.c.2-gp.log

@@ -17,4 +17,6 @@ gp> order = charorder(g,chi)
 4
 gp> [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 [[1]~, [3]~]
+gp> nfinit(polcyclo(charorder(g,chi)))
+[x^2 + 1, [0, 1], -4, 1, [Mat([1, 0.E-57 + 1.0000000000000000000000000000000000000*I]), [1, 1.0000000000000000000000000000000000000; 1, -1.0000000000000000000000000000000000000], [16, 16; 16, -16], [2, 0; 0, -2], [2, 0; 0, 2], [1, 0; 0, -1], [1, [0, -1; 1, 0]], [2]], [0.E-57 + 1.0000000000000000000000000000000000000*I], [1, x], [1, 0; 0, 1], [1, 0, 0, -1; 0, 1, 1, 0]]
 gp> 

lmfdb/tests/snippet_tests/characters/code-5.c.2-sage.log

@@ -18,4 +18,8 @@ True
 sage: chi.galois_orbit()
 [Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -zeta4,
  Dirichlet character modulo 5 of conductor 5 mapping 2 |--> zeta4]
+sage: CyclotomicField(chi.multiplicative_order())
+Cyclotomic Field of order 4 and degree 2
+sage: chi.fixed_field()
+Number Field in a with defining polynomial x^4 + x^3 + x^2 + x + 1
 sage: