more pep8 fixes in elliptic curves

Author: fchapoton

src/sage/schemes/elliptic_curves/cm.py

@@ -263,7 +263,7 @@ def is_HCP(f, check_monic_irreducible=True):
             continue
         if not fp.is_squarefree():
             continue
-        if d<h and d not in h2list:
+        if d < h and d not in h2list:
             return zero
         jp = fp.any_root(degree=-1, assume_squarefree=True)
         E = EllipticCurve(j=jp)
@@ -318,14 +318,15 @@ def OrderClassNumber(D0,h0,f):
     ps = f.prime_divisors()
     from sage.misc.misc_c import prod
     from sage.arith.misc import kronecker as kronecker_symbol
-    n = (f // prod(ps)) * prod(p-kronecker_symbol(D0,p) for p in ps)
+    n = (f // prod(ps)) * prod(p - kronecker_symbol(D0, p) for p in ps)
     if D0 == -3:
-        #assert h0 == 1 and n%3==0
-        return n//3
+        # assert h0 == 1 and n % 3 == 0
+        return n // 3
     if D0 == -4:
-        #assert h0 == 1 and n%2==0
-        return n//2
-    return n*h0
+        # assert h0 == 1 and n % 2 == 0
+        return n // 2
+    return n * h0
+
 
 @cached_function
 def cm_j_invariants(K, proof=None):
@@ -767,10 +768,10 @@ def discriminants_with_bounded_class_number(hmax, B=None, proof=None):
 
     # Easy case where we have already computed and cached the relevant values
     if hDf_dict and hmax <= max(hDf_dict):
-        T = {h:Dflist for h,Dflist in hDf_dict.items() if h<=hmax}
+        T = {h:Dflist for h,Dflist in hDf_dict.items() if h <= hmax}
         if B:
             for h in T:
-                T[h] = [Df for Df in T[h] if Df[0].abs()*Df[1]**2<=B]
+                T[h] = [Df for Df in T[h] if Df[0].abs()*Df[1]**2 <= B]
         return T
 
     # imports that are needed only for this function
@@ -821,7 +822,7 @@ def discriminants_with_bounded_class_number(hmax, B=None, proof=None):
         for D0,f in Dflist:
             h_dict[D0*f**2] = h
         if not count:
-            Dflist = [Df for Df in Dflist if Df[0].abs()*Df[1]**2<=B]
+            Dflist = [Df for Df in Dflist if Df[0].abs()*Df[1]**2 <= B]
         T[h] = set(Dflist)
 
     # We do not need to certify the class number from :pari:`qfbclassno` for discriminants under 2*10^10
@@ -993,7 +994,7 @@ def is_cm_j_invariant(j, algorithm='CremonaSutherland', method=None):
 
     h = jpol.degree()
     if algorithm in ['exhaustive', 'old']:
-        if h>100:
+        if h > 100:
             raise NotImplementedError("CM data only available for class numbers up to 100")
         for d,f in cm_orders(h):
             if jpol == hilbert_class_polynomial(d*f**2):
@@ -1046,8 +1047,8 @@ def is_cm_j_invariant(j, algorithm='CremonaSutherland', method=None):
             if cmd: # we have a candidate CM field already
                 break
             else:   # we need to try more primes
-                max_primes *=2
-        if D.valuation(P)>0: # skip bad primes
+                max_primes *= 2
+        if D.valuation(P) > 0: # skip bad primes
             continue
         aP = E.reduction(P).trace_of_frobenius()
         if aP == 0: # skip supersingular primes

src/sage/schemes/elliptic_curves/constructor.py

@@ -696,14 +696,14 @@ def coefficients_from_j(j, minimal_twist=True):
     if K not in _Fields:
         K = K.fraction_field()
 
-    char=K.characteristic()
-    if char==2:
+    char = K.characteristic()
+    if char == 2:
         if j == 0:
             return Sequence([0, 0, 1, 0, 0], universe=K)
         else:
             return Sequence([1, 0, 0, 0, 1/j], universe=K)
     if char == 3:
-        if j==0:
+        if j == 0:
             return Sequence([0, 0, 0, 1, 0], universe=K)
         else:
             return Sequence([0, j, 0, 0, -j**2], universe=K)
@@ -717,7 +717,7 @@ def coefficients_from_j(j, minimal_twist=True):
             return Sequence([0, 0, 0, -1, 0], universe=K) # 32a2
 
         if not minimal_twist:
-            k=j-1728
+            k = j-1728
             return Sequence([0, 0, 0, -3*j*k, -2*j*k**2], universe=K)
 
         n = j.numerator()
@@ -729,7 +729,7 @@ def coefficients_from_j(j, minimal_twist=True):
         from sage.sets.set import Set
         for p in Set(n.prime_divisors()+m.prime_divisors()):
             e = min(a4.valuation(p)//2,a6.valuation(p)//3)
-            if e>0:
+            if e > 0:
                 p = p**e
                 a4 /= p**2
                 a6 /= p**3
@@ -754,7 +754,7 @@ def coefficients_from_j(j, minimal_twist=True):
         return Sequence([0, 0, 0, 0, 1], universe=K)
     if j == 1728:
         return Sequence([0, 0, 0, 1, 0], universe=K)
-    k=j-1728
+    k = j-1728
     return Sequence([0, 0, 0, -3*j*k, -2*j*k**2], universe=K)
 
 
@@ -1112,7 +1112,7 @@ def EllipticCurve_from_cubic(F, P=None, morphism=True):
     # Test whether P is a flex; if not test whether there are any rational flexes:
 
     hessian = Matrix([[F.derivative(v1, v2) for v1 in R.gens()] for v2 in R.gens()]).det()
-    if P and hessian(P)==0:
+    if P and hessian(P) == 0:
         flex_point = P
     else:
         flexes = C.intersection(Curve(hessian)).rational_points()
@@ -1174,11 +1174,11 @@ def EllipticCurve_from_cubic(F, P=None, morphism=True):
         if not P:
             raise ValueError('A point must be given when the cubic has no rational flexes')
         L = tangent_at_smooth_point(C,P)
-        Qlist = [Q for Q in C.intersection(Curve(L)).rational_points() if C(Q)!=CP]
+        Qlist = [Q for Q in C.intersection(Curve(L)).rational_points() if C(Q) != CP]
         # assert Qlist
         P2 = C(Qlist[0])
         L2 = tangent_at_smooth_point(C,P2)
-        Qlist = [Q for Q in C.intersection(Curve(L2)).rational_points() if C(Q)!=P2]
+        Qlist = [Q for Q in C.intersection(Curve(L2)).rational_points() if C(Q) != P2]
         # assert Qlist
         P3 = C(Qlist[0])
 
@@ -1345,7 +1345,7 @@ def chord_and_tangent(F, P):
         raise TypeError('{} does not define a point on a projective curve over {} defined by {}'.format(P,K,F))
 
     L = Curve(tangent_at_smooth_point(C,P))
-    Qlist = [Q for Q in C.intersection(L).rational_points() if Q!=P]
+    Qlist = [Q for Q in C.intersection(L).rational_points() if Q != P]
     if Qlist:
         return Qlist[0]
     return P

src/sage/schemes/elliptic_curves/ec_database.py

@@ -133,7 +133,7 @@ def rank(self, rank, tors=0, n=10, labels=False):
             []
         """
         from sage.env import ELLCURVE_DATA_DIR
-        data = os.path.join(ELLCURVE_DATA_DIR, 'rank%s' %rank)
+        data = os.path.join(ELLCURVE_DATA_DIR, 'rank%s' % rank)
         try:
             f = open(data)
         except IOError:
@@ -151,7 +151,7 @@ def rank(self, rank, tors=0, n=10, labels=False):
             # NOTE: only change this bound below after checking/fixing
             # the Cremona labels in the elliptic_curves package!
             if N <= 400000:
-                label = '%s%s%s' %(N, iso, num)
+                label = '%s%s%s' % (N, iso, num)
             else:
                 label = None
 

src/sage/schemes/elliptic_curves/ell_field.py

@@ -155,7 +155,7 @@ def quadratic_twist(self, D=None):
                     # degree is odd we can take D=1; otherwise it suffices to
                     # consider odd powers of a generator.
                     D = K(1)
-                    if K.degree() %2==0:
+                    if K.degree() % 2 == 0:
                         D = K.gen()
                         a = D**2
                         while (x**2 + x + D).roots():
@@ -172,26 +172,26 @@ def quadratic_twist(self, D=None):
                 raise ValueError("twisting parameter D must be specified over infinite fields.")
         else:
             try:
-                D=K(D)
+                D = K(D)
             except ValueError:
                 raise ValueError("twisting parameter D must be in the base field.")
 
-            if char!=2 and D.is_zero():
+            if char != 2 and D.is_zero():
                 raise ValueError("twisting parameter D must be nonzero when characteristic is not 2")
 
-        if char!=2:
-            b2,b4,b6,b8=self.b_invariants()
+        if char != 2:
+            b2,b4,b6,b8 = self.b_invariants()
             # E is isomorphic to  [0,b2,0,8*b4,16*b6]
             return EllipticCurve(K,[0,b2*D,0,8*b4*D**2,16*b6*D**3])
 
         # now char==2
-        if self.j_invariant() !=0: # iff a1!=0
-            a1,a2,a3,a4,a6=self.ainvs()
-            E0=self.change_weierstrass_model(a1,a3/a1,0,(a1**2*a4+a3**2)/a1**3)
+        if self.j_invariant() != 0: # iff a1!=0
+            a1,a2,a3,a4,a6 = self.ainvs()
+            E0 = self.change_weierstrass_model(a1,a3/a1,0,(a1**2*a4+a3**2)/a1**3)
             # which has the form = [1,A2,0,0,A6]
-            assert E0.a1()==K(1)
-            assert E0.a3()==K(0)
-            assert E0.a4()==K(0)
+            assert E0.a1() == K(1)
+            assert E0.a3() == K(0)
+            assert E0.a4() == K(0)
             return EllipticCurve(K,[1,E0.a2()+D,0,0,E0.a6()])
         else:
             raise ValueError("Quadratic twist not implemented in char 2 when j=0")
@@ -229,8 +229,8 @@ def two_torsion_rank(self):
             sage: EllipticCurve('15a1').two_torsion_rank()
             2
         """
-        f=self.division_polynomial(rings.Integer(2))
-        n=len(f.roots())+1
+        f = self.division_polynomial(rings.Integer(2))
+        n = len(f.roots())+1
         return rings.Integer(n).ord(rings.Integer(2))
 
     def quartic_twist(self, D):
@@ -258,22 +258,22 @@ def quartic_twist(self, D):
             sage: E.is_isomorphic(E1, GF(13^4,'a'))                                     # optional - sage.rings.finite_rings
             True
         """
-        K=self.base_ring()
-        char=K.characteristic()
-        D=K(D)
+        K = self.base_ring()
+        char = K.characteristic()
+        D = K(D)
 
-        if char==2 or char==3:
+        if char == 2 or char == 3:
             raise ValueError("Quartic twist not defined in chars 2,3")
 
-        if self.j_invariant() !=K(1728):
+        if self.j_invariant() != K(1728):
             raise ValueError("Quartic twist not defined when j!=1728")
 
         if D.is_zero():
             raise ValueError("quartic twist requires a nonzero argument")
 
-        c4,c6=self.c_invariants()
+        c4,c6 = self.c_invariants()
         # E is isomorphic to  [0,0,0,-27*c4,0]
-        assert c6==0
+        assert c6 == 0
         return EllipticCurve(K,[0,0,0,-27*c4*D,0])
 
     def sextic_twist(self, D):
@@ -303,22 +303,22 @@ def sextic_twist(self, D):
             sage: E.is_isomorphic(E1, GF(13^6,'a'))                                     # optional - sage.rings.finite_rings
             True
         """
-        K=self.base_ring()
-        char=K.characteristic()
-        D=K(D)
+        K = self.base_ring()
+        char = K.characteristic()
+        D = K(D)
 
-        if char==2 or char==3:
+        if char == 2 or char == 3:
             raise ValueError("Sextic twist not defined in chars 2,3")
 
-        if self.j_invariant() !=K(0):
+        if self.j_invariant() != K(0):
             raise ValueError("Sextic twist not defined when j!=0")
 
         if D.is_zero():
             raise ValueError("Sextic twist requires a nonzero argument")
 
-        c4,c6=self.c_invariants()
+        c4,c6 = self.c_invariants()
         # E is isomorphic to  [0,0,0,0,-54*c6]
-        assert c4==0
+        assert c4 == 0
         return EllipticCurve(K,[0,0,0,0,-54*c6*D])
 
     def is_quadratic_twist(self, other):
@@ -420,7 +420,7 @@ def is_quadratic_twist(self, other):
         zero = K.zero()
         if not K == F.base_ring():
             return zero
-        j=E.j_invariant()
+        j = E.j_invariant()
         if j != F.j_invariant():
             return zero
 
@@ -429,12 +429,12 @@ def is_quadratic_twist(self, other):
                 return rings.ZZ(1)
             return K.one()
 
-        char=K.characteristic()
+        char = K.characteristic()
 
-        if char==2:
+        if char == 2:
             raise NotImplementedError("not implemented in characteristic 2")
-        elif char==3:
-            if j==0:
+        elif char == 3:
+            if j == 0:
                 raise NotImplementedError("not implemented in characteristic 3 for curves of j-invariant 0")
             D = E.b2()/F.b2()
 
@@ -443,18 +443,18 @@ def is_quadratic_twist(self, other):
             c4E,c6E = E.c_invariants()
             c4F,c6F = F.c_invariants()
 
-            if j==0:
+            if j == 0:
                 um = c6E/c6F
-                x=rings.polygen(K)
-                ulist=(x**3-um).roots(multiplicities=False)
+                x = rings.polygen(K)
+                ulist = (x**3-um).roots(multiplicities=False)
                 if not ulist:
                     D = zero
                 else:
                     D = ulist[0]
-            elif j==1728:
-                um=c4E/c4F
-                x=rings.polygen(K)
-                ulist=(x**2-um).roots(multiplicities=False)
+            elif j == 1728:
+                um = c4E/c4F
+                x = rings.polygen(K)
+                ulist = (x**2-um).roots(multiplicities=False)
                 if not ulist:
                     D = zero
                 else:
@@ -519,18 +519,18 @@ def is_quartic_twist(self, other):
         zero = K.zero()
         if not K == F.base_ring():
             return zero
-        j=E.j_invariant()
-        if j != F.j_invariant() or j!=K(1728):
+        j = E.j_invariant()
+        if j != F.j_invariant() or j != K(1728):
             return zero
 
         if E.is_isomorphic(F):
             return K.one()
 
-        char=K.characteristic()
+        char = K.characteristic()
 
-        if char==2:
+        if char == 2:
             raise NotImplementedError("not implemented in characteristic 2")
-        elif char==3:
+        elif char == 3:
             raise NotImplementedError("not implemented in characteristic 3")
         else:
             # now char!=2,3:
@@ -588,18 +588,18 @@ def is_sextic_twist(self, other):
         zero = K.zero()
         if not K == F.base_ring():
             return zero
-        j=E.j_invariant()
+        j = E.j_invariant()
         if j != F.j_invariant() or not j.is_zero():
             return zero
 
         if E.is_isomorphic(F):
             return K.one()
 
-        char=K.characteristic()
+        char = K.characteristic()
 
-        if char==2:
+        if char == 2:
             raise NotImplementedError("not implemented in characteristic 2")
-        elif char==3:
+        elif char == 3:
             raise NotImplementedError("not implemented in characteristic 3")
         else:
             # now char!=2,3:
@@ -1757,7 +1757,7 @@ def hasse_invariant(self):
             R = k['x']
             x = R.gen()
             E = self.short_weierstrass_model()
-            f=(x**3+E.a4()*x+E.a6())**((p-1)//2)
+            f = (x**3+E.a4()*x+E.a6())**((p-1)//2)
             return f.coefficients(sparse=False)[p-1]
 
     def isogeny_ell_graph(self, l, directed=True, label_by_j=False):