Some pep8 in elliptic curves

src/sage/schemes/elliptic_curves/cm.py

@@ -157,7 +157,7 @@ def hilbert_class_polynomial(D, algorithm=None):
     h = len(rqf) # class number
     c1 = 3.05682737291380 # log(2*10.63)
     c2 = sum([1/RR(qf[0]) for qf in rqf], RR(0))
-    prec =  c2*RR(3.142)*RR(D).abs().sqrt() + h*c1  # bound on log
+    prec = c2 * RR(3.142) * RR(D).abs().sqrt() + h * c1  # bound on log
     prec = prec * 1.45   # bound on log_2 (1/log(2) = 1.44..)
     prec = 10 + prec.ceil()  # allow for rounding error
 
@@ -242,7 +242,8 @@ def is_HCP(f, check_monic_irreducible=True):
     from sage.rings.finite_rings.finite_field_constructor import GF
 
     h = f.degree()
-    h2list = [d for d in h.divisors() if (d-h)%2 == 0 and d.prime_to_m_part(2) == 1]
+    h2list = [d for d in h.divisors()
+              if (d-h) % 2 == 0 and d.prime_to_m_part(2) == 1]
     pmin = 33 * (h**2 * (RR(h+2).log().log()+2)**2).ceil()
     # Guarantees 4*p > |D| for fundamental D under GRH
     p = pmin-1
@@ -262,7 +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)
@@ -317,14 +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):
@@ -766,10 +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
@@ -820,7 +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
@@ -831,7 +833,7 @@ def discriminants_with_bounded_class_number(hmax, B=None, proof=None):
         if not D.is_discriminant():
             continue
         D0 = D.squarefree_part()
-        if D0%4 !=1:
+        if D0 % 4 != 1:
             D0 *= 4
         f = (D//D0).isqrt()
 
@@ -983,16 +985,16 @@ def is_cm_j_invariant(j, algorithm='CremonaSutherland', method=None):
         D = is_HCP(jpol, check_monic_irreducible=False)
         if D:
             D0 = D.squarefree_part()
-            if D0%4 !=1:
+            if D0 % 4 != 1:
                 D0 *= 4
-            f = ZZ(D//D0).isqrt()
-            return (True, (D0,f))
+            f = ZZ(D // D0).isqrt()
+            return (True, (D0, f))
         else:
             return (False, 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):
@@ -1020,8 +1022,8 @@ def is_cm_j_invariant(j, algorithm='CremonaSutherland', method=None):
     from sage.schemes.elliptic_curves.constructor import EllipticCurve
     E = EllipticCurve(j=j).integral_model()
     D = E.discriminant()
-    prime_bound = 1000 # test primes of degree 1 up to this norm
-    max_primes =    20 # test at most this many primes
+    prime_bound = 1000  # test primes of degree 1 up to this norm
+    max_primes = 20     # test at most this many primes
     num_prime = 0
     cmd = 0
     cmf = 0
@@ -1045,8 +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_curve_isogeny.py

@@ -483,13 +483,13 @@ def compute_codomain_kohel(E, kernel):
 
             v, w = compute_vw_kohel_even_deg1(x0, y0, a1, a2, a4)
 
-        elif n == 3: # psi_2tor is the full 2-division polynomial
+        elif n == 3:  # psi_2tor is the full 2-division polynomial
 
             b2, b4, _, _ = E.b_invariants()
 
-            s1 = -psi_2tor[n-1]
-            s2 =  psi_2tor[n-2]
-            s3 = -psi_2tor[n-3]
+            s1 = -psi_2tor[n - 1]
+            s2 = psi_2tor[n - 2]
+            s3 = -psi_2tor[n - 3]
 
             v, w = compute_vw_kohel_even_deg3(b2, b4, s1, s2, s3)
 
@@ -499,9 +499,9 @@ def compute_codomain_kohel(E, kernel):
 
         b2, b4, b6, _ = E.b_invariants()
 
-        s1 = -psi[n-1] if n >= 1 else 0
-        s2 =  psi[n-2] if n >= 2 else 0
-        s3 = -psi[n-3] if n >= 3 else 0
+        s1 = -psi[n - 1] if n >= 1 else 0
+        s2 = psi[n - 2] if n >= 2 else 0
+        s3 = -psi[n - 3] if n >= 3 else 0
 
         # initializing these allows us to calculate E2.
         v, w = compute_vw_kohel_odd(b2, b4, b6, s1, s2, s3, n)
@@ -1893,7 +1893,7 @@ def __init_from_kernel_list(self, kernel_gens):
                to Elliptic Curve defined by y^2 = x^3 + 80816485163488178037199320944019099858815874115367810482828676054000067654558381377552245721755005198633191074893*x + 301497584865165444049833326660609767433467459033532853758006118022998267706948164646650354324860226263546558337993
                   over Finite Field of size 461742260113997803268895001173557974278278194575766957660028841364655249961609425998827452443620996655395008156411
         """
-        if self.__check :
+        if self.__check:
             for P in kernel_gens:
                 if not P.has_finite_order():
                     raise ValueError("given kernel contains point of infinite order")
@@ -2337,7 +2337,7 @@ def __init_even_kernel_polynomial(self, E, psi_G):
             (x^7 + 5*x^6 + 2*x^5 + 6*x^4 + 3*x^3 + 5*x^2 + 6*x + 3, (x^9 + 4*x^8 + 2*x^7 + 4*x^3 + 2*x^2 + x + 6)*y, 1, 6, 3, 4)
         """
         # check if the polynomial really divides the two_torsion_polynomial
-        if self.__check and E.division_polynomial(2, x=self.__poly_ring.gen()) % psi_G != 0 :
+        if self.__check and E.division_polynomial(2, x=self.__poly_ring.gen()) % psi_G != 0:
             raise ValueError(f"the polynomial {psi_G} does not define a finite subgroup of {E}")
 
         n = psi_G.degree() # 1 or 3
@@ -2363,9 +2363,9 @@ def __init_even_kernel_polynomial(self, E, psi_G):
             omega = (y*psi_G**2 - v*(a1*psi_G + (y - y0)))*psi_G
 
         elif n == 3:
-            s1 = -psi_G[n-1]
-            s2 =  psi_G[n-2]
-            s3 = -psi_G[n-3]
+            s1 = -psi_G[n - 1]
+            s2 = psi_G[n - 2]
+            s3 = -psi_G[n - 3]
 
             psi_G_pr = psi_G.derivative()
             psi_G_prpr = psi_G_pr.derivative()