Trac #34644: minor details in schemes

some about E241 and E502

URL: https://trac.sagemath.org/34644
Reported by: chapoton
Ticket author(s): Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

src/sage/schemes/elliptic_curves/ell_generic.py

@@ -1918,9 +1918,9 @@ def division_polynomial(self, m, x=None, two_torsion_multiplicity=2, force_evalu
                     and x.lift().base_ring() is self.base_ring():
                 d = x.parent().modulus().degree()
                 evaluate = m < 220 or \
-                            (d <  10 and m < 420) or (d <  15 and m < 340) or \
-                            (d <  30 and m < 280) or (d < 100 and m < 250) or \
-                            m <= min(250, d)
+                    (d < 10 and m < 420) or (d < 15 and m < 340) or \
+                    (d < 30 and m < 280) or (d < 100 and m < 250) or \
+                    m <= min(250, d)
 
         # Check if we should (attempt to) compute the result by simply
         # evaluating a cached polynomial at the given input.
@@ -2849,16 +2849,16 @@ def montgomery_model(self, twisted=False, morphism=False):
                 return E, self.isomorphism_to(E)
             return E
 
-        P2, (x,y,z) = self.ambient_space().objgens()
+        P2, (x, y, z) = self.ambient_space().objgens()
         f = B * y**2*z - x * (x * (x + A*z) + z**2)
         C = plane_curve.ProjectivePlaneCurve(P2, f)
 
         if not morphism:
             return C
 
         t = ~(B * s).sqrt()
-        iso_maps = (x     - r * z,  t * y  ,  s * z)
-        inv_maps = (x * s + r * z,  s * y/t,      z)
+        iso_maps = (x - r * z, t * y, s * z)
+        inv_maps = (x * s + r * z, s * y / t, z)
 
         w = self.isomorphism_to(Ew)
         wmap, winv = w.rational_maps(), (~w).rational_maps()
@@ -2868,7 +2868,7 @@ def montgomery_model(self, twisted=False, morphism=False):
         inv = [f(*inv_maps) for f in winv]
 
         from sage.schemes.elliptic_curves.weierstrass_transform \
-                import WeierstrassTransformationWithInverse as WTI
+            import WeierstrassTransformationWithInverse as WTI
         iso = WTI(self, C, iso, 1, inv, s**-3)
         return C, iso
 
@@ -3194,7 +3194,7 @@ def _p_primary_torsion_basis(self, p, m=None):
         k = 1
         log_order = 2
         if m <= log_order:
-            return [[P1,1],[P2,1]]
+            return [[P1, 1], [P2, 1]]
 
         pts1 = P1.division_points(p)
         pts2 = P2.division_points(p)
@@ -3255,7 +3255,7 @@ def _p_primary_torsion_basis(self, p, m=None):
                 return [[P1, n], [P2, k]]
             pts = P1.division_points(p)
             if not pts:
-                for Q in generic.multiples(P2,p-1,P1+P2,operation='+'):
+                for Q in generic.multiples(P2, p-1, P1+P2, operation='+'):
                     # Q runs through P1+a*P2 for a=1,2,...,p-1
                     pts = Q.division_points(p)
                     if pts:

src/sage/schemes/elliptic_curves/heegner.py

@@ -117,15 +117,15 @@
 from sage.modular.modsym.p1list import P1List
 
 
-##################################################################################
+###############################################################################
 #
 # The exported functions, which are in most cases enough to get the
 # user going working with Heegner points:
 #
 #    heegner_points -- all of them with given level, discriminant, conductor
 #    heegner_point -- a specific one
 #
-##################################################################################
+###############################################################################
 
 def heegner_points(N, D=None, c=None):
     """
@@ -135,11 +135,11 @@ def heegner_points(N, D=None, c=None):
 
     INPUT:
 
-        - `N` -- level (positive integer)
+    - `N` -- level (positive integer)
 
-        - `D` -- discriminant (negative integer)
+    - `D` -- discriminant (negative integer)
 
-        - `c` -- conductor (positive integer)
+    - `c` -- conductor (positive integer)
 
     EXAMPLES::
 
@@ -155,9 +155,10 @@ def heegner_points(N, D=None, c=None):
     if D is not None and c is None:
         return HeegnerPoints_level_disc(N, D)
     if D is not None and c is not None:
-        return HeegnerPoints_level_disc_cond(N,D,c)
+        return HeegnerPoints_level_disc_cond(N, D, c)
     raise TypeError
 
+
 def heegner_point(N, D=None, c=1):
     """
     Return a specific Heegner point of level `N` with given
@@ -167,11 +168,11 @@ def heegner_point(N, D=None, c=1):
 
     INPUT:
 
-        - `N` -- level (positive integer)
+    - `N` -- level (positive integer)
 
-        - `D` -- discriminant (optional: default first valid `D`)
+    - `D` -- discriminant (optional: default first valid `D`)
 
-        - `c` -- conductor (positive integer, optional, default: 1)
+    - `c` -- conductor (positive integer, optional, default: 1)
 
     EXAMPLES::
 
@@ -185,20 +186,20 @@ def heegner_point(N, D=None, c=1):
         Heegner point 1/778*sqrt(-20) - 165/389 of discriminant -20 on X_0(389)
     """
     if D is not None:
-        return heegner_points(N,D,c)[0]
+        return heegner_points(N, D, c)[0]
     H = heegner_points(N)
     D = H.discriminants(1)[0]
-    return heegner_points(N,D,c)[0]
+    return heegner_points(N, D, c)[0]
 
 
-##################################################################################
+###############################################################################
 #
 # Ring class fields, represented as abstract objects.  These do not
 # derive from number fields, since we do not need to work with their
 # elements, and explicitly representing them as number fields would be
 # far too difficult.
 #
-##################################################################################
+###############################################################################
 
 class RingClassField(SageObject):
     """
@@ -352,9 +353,8 @@ def _repr_(self):
         """
         c = self.__c
         if c == 1:
-            return "Hilbert class field of QQ[sqrt(%s)]"%self.__D
-        else:
-            return "Ring class field extension of QQ[sqrt(%s)] of conductor %s"%(self.__D, self.__c)
+            return "Hilbert class field of QQ[sqrt(%s)]" % self.__D
+        return "Ring class field extension of QQ[sqrt(%s)] of conductor %s" % (self.__D, self.__c)
 
     @cached_method
     def degree_over_K(self):
@@ -6533,7 +6533,8 @@ def heegner_point_height(self, D, prec=2, check_rank=True):
         return IR(alpha-MIN_ERR,alpha+MIN_ERR) * IR(LE1-err_E,LE1+err_E) * IR(LF1-err_F,LF1+err_F)
 
 
-def heegner_index(self, D,  min_p=2, prec=5, descent_second_limit=12, verbose_mwrank=False, check_rank=True):
+def heegner_index(self, D, min_p=2, prec=5, descent_second_limit=12,
+                  verbose_mwrank=False, check_rank=True):
     r"""
     Return an interval that contains the index of the Heegner
     point `y_K` in the group of `K`-rational points modulo torsion
@@ -6745,7 +6746,7 @@ def _adjust_heegner_index(self, a):
     return a.sqrt()
 
 
-def heegner_index_bound(self, D=0,  prec=5, max_height=None):
+def heegner_index_bound(self, D=0, prec=5, max_height=None):
     r"""
     Assume ``self`` has rank 0.
 

src/sage/schemes/hyperelliptic_curves/constructor.py

@@ -6,15 +6,13 @@
 - David Kohel (2006): initial version
 
 - Anna Somoza (2019-04): dynamic class creation
-
 """
-
-#*****************************************************************************
+# ****************************************************************************
 #  Copyright (C) 2006 David Kohel <[email protected]>
 #                2019 Anna Somoza <[email protected]>
 #  Distributed under the terms of the GNU General Public License (GPL)
-#                  http://www.gnu.org/licenses/
-#*****************************************************************************
+#                  https://www.gnu.org/licenses/
+# ****************************************************************************
 
 from sage.schemes.projective.projective_space import ProjectiveSpace
 
@@ -31,6 +29,7 @@
 
 from sage.structure.dynamic_class import dynamic_class
 
+
 def HyperellipticCurve(f, h=0, names=None, PP=None, check_squarefree=True):
     r"""
     Returns the hyperelliptic curve `y^2 + h y = f`, for
@@ -216,40 +215,39 @@ def HyperellipticCurve(f, h=0, names=None, PP=None, check_squarefree=True):
         if P(2) == 0:
             # characteristic 2
             if h == 0:
-                raise ValueError("In characteristic 2, argument h (= %s) must be non-zero."%h)
+                raise ValueError("In characteristic 2, argument h (= %s) must be non-zero." % h)
             if h[g+1] == 0 and f[2*g+1]**2 == f[2*g+2]*h[g]**2:
-                raise ValueError("Not a hyperelliptic curve: " \
-                                  "highly singular at infinity.")
+                raise ValueError("Not a hyperelliptic curve: "
+                                 "highly singular at infinity.")
             should_be_coprime = [h, f*h.derivative()**2+f.derivative()**2]
         else:
             # characteristic not 2
             if not F.degree() in [2*g+1, 2*g+2]:
-                raise ValueError("Not a hyperelliptic curve: " \
-                                  "highly singular at infinity.")
+                raise ValueError("Not a hyperelliptic curve: "
+                                 "highly singular at infinity.")
             should_be_coprime = [F, F.derivative()]
         try:
-            smooth = should_be_coprime[0].gcd(should_be_coprime[1]).degree()==0
+            smooth = should_be_coprime[0].gcd(should_be_coprime[1]).degree() == 0
         except (AttributeError, NotImplementedError, TypeError):
             try:
-                smooth = should_be_coprime[0].resultant(should_be_coprime[1])!=0
+                smooth = should_be_coprime[0].resultant(should_be_coprime[1]) != 0
             except (AttributeError, NotImplementedError, TypeError):
-                raise NotImplementedError("Cannot determine whether " \
-                      "polynomials %s have a common root. Use " \
-                      "check_squarefree=False to skip this check." % \
+                raise NotImplementedError("Cannot determine whether "
+                      "polynomials %s have a common root. Use "
+                      "check_squarefree=False to skip this check." %
                       should_be_coprime)
         if not smooth:
-            raise ValueError("Not a hyperelliptic curve: " \
-                              "singularity in the provided affine patch.")
+            raise ValueError("Not a hyperelliptic curve: "
+                             "singularity in the provided affine patch.")
     R = P.base_ring()
     PP = ProjectiveSpace(2, R)
     if names is None:
-        names = ["x","y"]
+        names = ["x", "y"]
 
     superclass = []
     cls_name = ["HyperellipticCurve"]
 
-    genus_classes = {
-        2 :  HyperellipticCurve_g2}
+    genus_classes = {2: HyperellipticCurve_g2}
 
     is_pAdicField = lambda x: isinstance(x, sage.rings.abc.pAdicField)
 

src/sage/schemes/plane_conics/con_field.py

@@ -40,6 +40,7 @@
 from sage.categories.fields import Fields
 _Fields = Fields()
 
+
 class ProjectiveConic_field(ProjectivePlaneCurve_field):
     r"""
     Create a projective plane conic curve over a field.
@@ -113,10 +114,10 @@ def base_extend(self, S):
             if B == S:
                 return self
             if not S.has_coerce_map_from(B):
-                raise ValueError("No natural map from the base ring of self " \
+                raise ValueError("No natural map from the base ring of self "
                                   "(= %s) to S (= %s)" % (self, S))
             from .constructor import Conic
-            con = Conic([S(c) for c in self.coefficients()], \
+            con = Conic([S(c) for c in self.coefficients()],
                         self.variable_names())
             if self._rational_point is not None:
                 pt = [S(c) for c in Sequence(self._rational_point)]
@@ -188,9 +189,7 @@ def derivative_matrix(self):
             [1 2 1]
             [1 1 0]
 
-        An example in characteristic `2`:
-
-        ::
+        An example in characteristic `2`::
 
             sage: P.<t> = GF(2)[]
             sage: c = Conic([t, 1, t^2, 1, 1, 0]); c
@@ -203,9 +202,9 @@ def derivative_matrix(self):
             [t^2   1   0]
         """
         a, b, c, d, e, f = self.coefficients()
-        return Matrix([[ 2*a ,   b ,   c ],
-                       [   b , 2*d ,   e ],
-                       [   c ,   e , 2*f ]])
+        return Matrix([[2 * a, b, c],
+                       [b, 2 * d, e],
+                       [c, e, 2 * f]])
 
     def determinant(self):
         r"""
@@ -497,8 +496,8 @@ def has_rational_point(self, point=False,
             # writing) fraction field elements are not converted automatically
             # from Magma to Sage.
             try:
-                return True, self.point( \
-                  [B(c.Numerator().sage()/c.Denominator().sage()) for c in pt])
+                return True, self.point(
+                    [B(c.Numerator().sage() / c.Denominator().sage()) for c in pt])
             except (TypeError, NameError):
                 pass
 
@@ -539,8 +538,8 @@ def has_rational_point(self, point=False,
             if point:
                 return ret
             return ret[0]
-        raise NotImplementedError("has_rational_point not implemented for " \
-                                   "conics over base field %s" % B)
+        raise NotImplementedError("has_rational_point not implemented for "
+                                  "conics over base field %s" % B)
 
     def has_singular_point(self, point=False):
         r"""
@@ -693,8 +692,8 @@ def hom(self, x, Y=None):
             if Y is None:
                 Y = im
             elif not Y == im:
-                raise ValueError("The matrix x (= %s) does not define a " \
-                                 "map from self (= %s) to Y (= %s)" % \
+                raise ValueError("The matrix x (= %s) does not define a "
+                                 "map from self (= %s) to Y (= %s)" %
                                  (x, self, Y))
             x = Sequence(x*vector(self.ambient_space().gens()))
             return self.Hom(Y)(x, check=False)
@@ -1120,12 +1119,12 @@ def rational_point(self, algorithm='default', read_cache=True):
             ...
             ValueError: Conic Projective Conic Curve over Real Field with 53 bits of precision defined by x^2 + y^2 + z^2 has no rational points over Real Field with 53 bits of precision!
         """
-        bl,pt = self.has_rational_point(point=True, algorithm=algorithm,
-                                        read_cache=read_cache)
+        bl, pt = self.has_rational_point(point=True, algorithm=algorithm,
+                                         read_cache=read_cache)
         if bl:
             return pt
-        raise ValueError("Conic %s has no rational points over %s!" % \
-                          (self, self.ambient_space().base_ring()))
+        raise ValueError("Conic %s has no rational points over %s!" %
+                         (self, self.ambient_space().base_ring()))
 
     def singular_point(self):
         r"""
@@ -1149,8 +1148,8 @@ def singular_point(self):
         """
         b = self.has_singular_point(point=True)
         if not b[0]:
-            raise ValueError("The conic self (= %s) has no rational " \
-                              "singular point" % self)
+            raise ValueError("The conic self (= %s) has no rational "
+                             "singular point" % self)
         return b[1]
 
     def symmetric_matrix(self):
@@ -1175,13 +1174,13 @@ def symmetric_matrix(self):
         a, b, c, d, e, f = self.coefficients()
         if self.base_ring().characteristic() == 2:
             if b == 0 and c == 0 and e == 0:
-                return Matrix([[a,0,0],[0,d,0],[0,0,f]])
-            raise ValueError("The conic self (= %s) has no symmetric matrix " \
-                              "because the base field has characteristic 2" % \
-                              self)
-        return Matrix([[  a , b/2, c/2 ],
-                       [ b/2,  d , e/2 ],
-                       [ c/2, e/2,  f  ]])
+                return Matrix([[a, 0, 0], [0, d, 0], [0, 0, f]])
+            raise ValueError("The conic self (= %s) has no symmetric matrix "
+                             "because the base field has characteristic 2" %
+                             self)
+        return Matrix([[a, b / 2, c / 2],
+                       [b / 2, d, e / 2],
+                       [c / 2, e / 2, f]])
 
     def upper_triangular_matrix(self):
         r"""