some care for blank lines in pyx in rings

src/sage/rings/complex_arb.pyx

@@ -1261,6 +1261,7 @@ class ComplexBallField(UniqueRepresentation, sage.rings.abc.ComplexBallField):
 
         return res
 
+
 cdef inline bint _do_sig(long prec) noexcept:
     """
     Whether signal handlers should be installed for calls to FLINT.

src/sage/rings/complex_double.pyx

@@ -505,7 +505,6 @@ cdef class ComplexDoubleField_class(sage.rings.abc.ComplexDoubleField):
             from sage.rings.complex_mpfr import ComplexField
             return ComplexField(prec)
 
-
     def gen(self, n=0):
         """
         Return the generator of the complex double field.
@@ -699,6 +698,7 @@ cdef ComplexDoubleElement new_ComplexDoubleElement():
     z = ComplexDoubleElement.__new__(ComplexDoubleElement)
     return z
 
+
 def is_ComplexDoubleElement(x):
     """
     Return ``True`` if ``x`` is a :class:`ComplexDoubleElement`.
@@ -1530,7 +1530,6 @@ cdef class ComplexDoubleElement(FieldElement):
         else:
             return z
 
-
     def is_square(self):
         r"""
         This function always returns ``True`` as `\CC` is algebraically closed.
@@ -2018,7 +2017,6 @@ cdef class ComplexDoubleElement(FieldElement):
         """
         return self._new_c(gsl_complex_tanh(self._complex))
 
-
     def sech(self):
         r"""
         This function returns the complex hyperbolic secant of the complex
@@ -2594,8 +2592,6 @@ cdef class ComplexToCDF(Morphism):
         return "Native"
 
 
-
-
 #####################################################
 # unique objects
 #####################################################
@@ -2604,6 +2600,7 @@ _CDF = ComplexDoubleField_class()
 CDF = _CDF  # external interface
 cdef ComplexDoubleElement I = ComplexDoubleElement(0,1)
 
+
 def ComplexDoubleField():
     """
     Returns the field of double precision complex numbers.
@@ -2617,6 +2614,7 @@ def ComplexDoubleField():
     """
     return _CDF
 
+
 from sage.misc.parser import Parser
 cdef cdf_parser = Parser(float, float,  {"I" : _CDF.gen(), "i" : _CDF.gen()})
 

src/sage/rings/complex_mpc.pyx

@@ -95,6 +95,7 @@ AA = None
 QQbar = None
 CDF = CLF = RLF = None
 
+
 def late_import():
     """
     Import the objects/modules after build (when needed).
@@ -112,6 +113,7 @@ def late_import():
         from sage.rings.real_lazy import CLF, RLF
         from sage.rings.complex_double import CDF
 
+
 _mpfr_rounding_modes = ['RNDN', 'RNDZ', 'RNDU', 'RNDD']
 
 _mpc_rounding_modes = [ 'RNDNN', 'RNDZN', 'RNDUN', 'RNDDN',
@@ -233,6 +235,8 @@ cpdef inline split_complex_string(string, int base=10):
 # their parent via direct C calls, which will be faster.
 
 cache = {}
+
+
 def MPComplexField(prec=53, rnd="RNDNN", names=None):
     """
     Return the complex field with real and imaginary parts having
@@ -2412,6 +2416,7 @@ cdef inline mp_exp_t max_exp(MPComplexNumber z) noexcept:
         return mpfr_get_exp(z.value.im)
     return max_exp_t(mpfr_get_exp(z.value.re), mpfr_get_exp(z.value.im))
 
+
 def __create__MPComplexField_version0 (prec, rnd):
     """
     Create a :class:`MPComplexField`.
@@ -2424,6 +2429,7 @@ def __create__MPComplexField_version0 (prec, rnd):
     """
     return MPComplexField(prec, rnd)
 
+
 def __create__MPComplexNumber_version0 (parent, s, base=10):
     """
     Create a :class:`MPComplexNumber`.
@@ -2439,6 +2445,7 @@ def __create__MPComplexNumber_version0 (parent, s, base=10):
     """
     return MPComplexNumber(parent, s, base=base)
 
+
 # original version of the file had this with only 1 underscore - TCS
 __create_MPComplexNumber_version0 = __create__MPComplexNumber_version0
 

src/sage/rings/complex_mpfr.pyx

@@ -69,6 +69,8 @@ NumberFieldElement_quadratic = ()
 AA = None
 QQbar = None
 CDF = CLF = RLF = None
+
+
 def late_import():
     """
     Import the objects/modules after build (when needed).
@@ -89,12 +91,14 @@ def late_import():
         from sage.rings.real_lazy import CLF, RLF
         from sage.rings.complex_double import CDF
 
+
 cdef object numpy_complex_interface = {'typestr': '=c16'}
 cdef object numpy_object_interface = {'typestr': '|O'}
 
 cdef mpfr_rnd_t rnd
 rnd = MPFR_RNDN
 
+
 def set_global_complex_round_mode(n):
     """
     Set the global complex rounding mode.
@@ -111,6 +115,7 @@ def set_global_complex_round_mode(n):
     global rnd
     rnd = n
 
+
 def is_ComplexNumber(x):
     r"""
     Return ``True`` if ``x`` is a complex number. In particular, if ``x`` is
@@ -145,6 +150,8 @@ def is_ComplexNumber(x):
 
 
 cache = {}
+
+
 def ComplexField(prec=53, names=None):
     """
     Return the complex field with real and imaginary parts having prec
@@ -673,7 +680,6 @@ class ComplexField_class(sage.rings.abc.ComplexField):
         from sage.categories.pushout import AlgebraicClosureFunctor
         return (AlgebraicClosureFunctor(), self._real_field())
 
-
     def random_element(self, component_max=1, *args, **kwds):
         r"""
         Return a uniformly distributed random number inside a square
@@ -857,6 +863,7 @@ class ComplexField_class(sage.rings.abc.ComplexField):
         return Factorization([(R(gg).monic(), e) for gg, e in zip(*F)],
                              f.leading_coefficient())
 
+
 cdef class ComplexNumber(sage.structure.element.FieldElement):
     """
     A floating point approximation to a complex number using any
@@ -1429,7 +1436,6 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
         """
         return gmpy2.GMPy_MPC_From_mpfr(self.__re, self.__im)
 
-
     def _mpmath_(self, prec=None, rounding=None):
         """
         Return an mpmath version of ``self``.
@@ -2300,8 +2306,6 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
         mpfr_clear(ch)
         return z
 
-
-
     def eta(self, omit_frac=False):
         r"""
         Return the value of the Dedekind `\eta` function on ``self``,
@@ -2367,7 +2371,6 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
         except sage.libs.pari.all.PariError:
             raise ValueError("value must be in the upper half plane")
 
-
     def sin(self):
         """
         Return the sine of ``self``.
@@ -2461,7 +2464,6 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
         mpfr_clear(a)
         return z
 
-
     def tanh(self):
         """
         Return the hyperbolic tangent of ``self``.
@@ -2729,7 +2731,6 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
         mpfr_atan2(x.value, self.__im, self.__re, rnd)
         return x
 
-
     def arg(self):
         """
         See :meth:`argument`.
@@ -3080,7 +3081,6 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
         mpfr_clear(r)
         return zlist
 
-
     def is_square(self):
         r"""
         This function always returns true as `\CC` is algebraically closed.
@@ -3498,6 +3498,7 @@ cpdef int cmp_abs(ComplexNumber a, ComplexNumber b) noexcept:
 
     return res
 
+
 def _format_complex_number(real, imag, format_spec):
     """
     Construct a formatted string from real and imaginary parts.

src/sage/rings/factorint.pyx

@@ -208,6 +208,7 @@ cpdef factor_aurifeuillian(n, check=True):
             return F
     return [n]
 
+
 def factor_cunningham(m, proof=None):
     r"""
     Return factorization of ``self`` obtained using trial division
@@ -251,6 +252,7 @@ def factor_cunningham(m, proof=None):
     else:
         return IntegerFactorization(L)*n.factor(proof=proof)
 
+
 cpdef factor_trial_division(m, long limit=LONG_MAX):
     r"""
     Return partial factorization of ``self`` obtained using trial division

src/sage/rings/finite_rings/element_givaro.pyx

@@ -1740,6 +1740,7 @@ def unpickle_FiniteField_givaroElement(parent, int x):
     """
     return make_FiniteField_givaroElement(parent._cache, x)
 
+
 from sage.misc.persist import register_unpickle_override
 register_unpickle_override('sage.rings.finite_field_givaro', 'unpickle_FiniteField_givaroElement', unpickle_FiniteField_givaroElement)
 

src/sage/rings/finite_rings/element_ntl_gf2e.pyx

@@ -1314,5 +1314,6 @@ def unpickleFiniteField_ntl_gf2eElement(parent, elem):
     """
     return parent(elem)
 
+
 from sage.misc.persist import register_unpickle_override
 register_unpickle_override('sage.rings.finite_field_ntl_gf2e', 'unpickleFiniteField_ntl_gf2eElement', unpickleFiniteField_ntl_gf2eElement)

src/sage/rings/finite_rings/finite_field_base.pyx

@@ -277,7 +277,6 @@ cdef class FiniteField(Field):
         return "GF(%s,Variable=>symbol %s)" % (self.order(),
                                                self.variable_name())
 
-
     def _sage_input_(self, sib, coerced):
         r"""
         Produce an expression which will reproduce this value when evaluated.
@@ -1115,7 +1114,6 @@ cdef class FiniteField(Field):
         """
         return self.order() - 1
 
-
     def random_element(self, *args, **kwds):
         r"""
         A random element of the finite field.  Passes arguments to
@@ -2162,6 +2160,7 @@ cdef class FiniteField(Field):
         python_int = int.from_bytes(input_bytes, byteorder=byteorder)
         return self.from_integer(python_int)
 
+
 def unpickle_FiniteField_ext(_type, order, variable_name, modulus, kwargs):
     r"""
     Used to unpickle extensions of finite fields. Now superseded (hence no
@@ -2177,6 +2176,7 @@ def unpickle_FiniteField_prm(_type, order, variable_name, kwargs):
     """
     return _type(order, variable_name, **kwargs)
 
+
 register_unpickle_override(
     'sage.rings.ring', 'unpickle_FiniteField_prm', unpickle_FiniteField_prm)
 

src/sage/rings/finite_rings/integer_mod.pyx

@@ -163,6 +163,7 @@ mod = Mod
 register_unpickle_override('sage.rings.integer_mod', 'Mod', Mod)
 register_unpickle_override('sage.rings.integer_mod', 'mod', mod)
 
+
 def IntegerMod(parent, value):
     """
     Create an integer modulo `n` with the given parent.
@@ -3462,7 +3463,6 @@ cdef class IntegerMod_int64(IntegerMod_abstract):
         """
         return self._new_c((self.ivalue * (<IntegerMod_int64>right).ivalue) % self._modulus.int64)
 
-
     cpdef _div_(self, right):
         """
         EXAMPLES::
@@ -3995,6 +3995,7 @@ def square_root_mod_prime_power(IntegerMod_abstract a, p, e):
         x *= p**(val//2)
     return x
 
+
 cpdef square_root_mod_prime(IntegerMod_abstract a, p=None):
     r"""
     Calculates the square root of `a`, where `a` is an

src/sage/rings/finite_rings/residue_field.pyx

@@ -919,6 +919,7 @@ class ResidueField_generic(Field):
         """
         return 1 + hash(self.ideal())
 
+
 cdef class ReductionMap(Map):
     """
     A reduction map from a (subset) of a number field or function field to
@@ -1685,6 +1686,7 @@ cdef class LiftingMap(Section):
         """
         return "Lifting"
 
+
 class ResidueFiniteField_prime_modn(ResidueField_generic, FiniteField_prime_modn):
     """
     The class representing residue fields of number fields that have

src/sage/rings/function_field/element.pyx

@@ -86,6 +86,7 @@ def is_FunctionFieldElement(x):
     from sage.rings.function_field.function_field import is_FunctionField
     return is_FunctionField(x.parent())
 
+
 def make_FunctionFieldElement(parent, element_class, representing_element):
     """
     Used for unpickling FunctionFieldElement objects (and subclasses).

src/sage/rings/function_field/hermite_form_polynomial.pyx

@@ -49,13 +49,14 @@ AUTHORS:
 #  Distributed under the terms of the GNU General Public License (GPL)
 #  as published by the Free Software Foundation; either version 2 of
 #  the License, or (at your option) any later version.
-#                  http://www.gnu.org/licenses/
+#                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
 from sage.matrix.matrix cimport Matrix
 from sage.rings.polynomial.polynomial_element cimport Polynomial
 from sage.matrix.constructor import identity_matrix
 
+
 def reversed_hermite_form(Matrix mat, bint transformation=False):
     """
     Transform the matrix in place to reversed hermite normal form and

src/sage/rings/integer_ring.pyx

@@ -87,6 +87,7 @@ cdef int number_of_integer_rings = 0
 #   sigma, we do not recreate the sampler but take it from this "cache".
 _prev_discrete_gaussian_integer_sampler = (None, None)
 
+
 def is_IntegerRing(x):
     r"""
     Internal function: return ``True`` iff ``x`` is the ring `\ZZ` of integers.
@@ -105,6 +106,7 @@ def is_IntegerRing(x):
     """
     return isinstance(x, IntegerRing_class)
 
+
 cdef class IntegerRing_class(CommutativeRing):
     r"""
     The ring of integers.
@@ -1406,7 +1408,6 @@ cdef class IntegerRing_class(CommutativeRing):
 
         g = g.gcd(R( {e[j] - e[i_min]: c[j] for j in range(i_min, k)} ))
 
-
         cdef list cc
         cdef list ee
         cdef int m1, m2
@@ -1609,6 +1610,7 @@ cdef class IntegerRing_class(CommutativeRing):
 ZZ = IntegerRing_class()
 Z = ZZ
 
+
 def IntegerRing():
     """
     Return the integer ring.
@@ -1622,6 +1624,7 @@ def IntegerRing():
     """
     return ZZ
 
+
 def crt_basis(X, xgcd=None):
     r"""
     Compute and return a Chinese Remainder Theorem basis for the list ``X``