sage.schemes: Update # needs

src/doc/en/reference/curves/index.rst

@@ -10,6 +10,7 @@ Curves
    sage/schemes/curves/projective_curve
    sage/schemes/curves/point
    sage/schemes/curves/closed_point
+   sage/schemes/curves/zariski_vankampen
 
    sage/schemes/jacobians/abstract_jacobian
 

src/sage/groups/finitely_presented.py

@@ -135,6 +135,7 @@
 from sage.libs.gap.libgap import libgap
 from sage.libs.gap.element import GapElement
 from sage.misc.cachefunc import cached_method
+from sage.groups.free_group import FreeGroup
 from sage.groups.free_group import FreeGroupElement
 from sage.functions.generalized import sign
 from sage.matrix.constructor import matrix
@@ -1426,6 +1427,35 @@ def simplified(self):
         """
         return self.simplification_isomorphism().codomain()
 
+    def sorted_presentation(self):
+        """
+        Return the same presentation with the relations sorted to ensure equality.
+
+        OUTPUT:
+
+        A new finitely presented group with the relations sorted.
+
+        EXAMPLES::
+
+            sage: G = FreeGroup(2) / [(1, 2, -1, -2), ()]; G
+            Finitely presented group < x0, x1 | x0*x1*x0^-1*x1^-1, 1 >
+            sage: G.sorted_presentation()
+            Finitely presented group < x0, x1 | 1, x1^-1*x0^-1*x1*x0 >
+        """
+        F = FreeGroup(self.ngens())
+        L0 = [r.Tietze() for r in self.relations()]
+        L1 = []
+        for rel in L0:
+            C = [rel]
+            for j in range(len(rel) - 1):
+                C.append(rel[j + 1:] + rel[:j + 1])
+            C1 = [tuple(-j for j in reversed(l)) for l in C]
+            C += C1
+            C.sort()
+            L1.append(C[0])
+        L1.sort()
+        return F/L1
+
     def epimorphisms(self, H):
         r"""
         Return the epimorphisms from `self` to `H`, up to automorphism of `H`.

src/sage/modular/abvar/homspace.py

@@ -148,14 +148,15 @@
 
 ::
 
-    sage: T = E.image_of_hecke_algebra()  # long time
-    sage: T.gens()  # long time
+    sage: # long time
+    sage: T = E.image_of_hecke_algebra()
+    sage: T.gens()
     (Abelian variety endomorphism of Abelian variety J0(33) of dimension 3,
      Abelian variety endomorphism of Abelian variety J0(33) of dimension 3,
      Abelian variety endomorphism of Abelian variety J0(33) of dimension 3)
-    sage: T.index_in(E)  # long time
+    sage: T.index_in(E)
     +Infinity
-    sage: T.index_in_saturation()  # long time
+    sage: T.index_in_saturation()
     1
 
 AUTHORS:

src/sage/modular/cusps_nf.py

@@ -1,3 +1,4 @@
+# sage.doctest: needs sage.rings.number_field
 r"""
 The set `\mathbb{P}^1(K)` of cusps of a number field `K`
 
@@ -62,7 +63,7 @@
 
     sage: Gamma0_NFCusps(N)
     [Cusp [0: 1] of Number Field in a with defining polynomial x^2 + 5,
-    Cusp [1: 3] of Number Field in a with defining polynomial x^2 + 5,
+     Cusp [1: 3] of Number Field in a with defining polynomial x^2 + 5,
     ...]
 """
 # ****************************************************************************
@@ -859,7 +860,7 @@ def ABmatrix(self):
             sage: M = alpha.ABmatrix()
             sage: M # random
             [-a^2 - a - 1, -3*a - 7, 8, -2*a^2 - 3*a + 4]
-            sage: M[0] == alpha.numerator() and M[2]==alpha.denominator()
+            sage: M[0] == alpha.numerator() and M[2] == alpha.denominator()
             True
 
         An AB-matrix associated to a cusp alpha will send Infinity to alpha:
@@ -870,7 +871,7 @@ def ABmatrix(self):
             sage: M = alpha.ABmatrix()
             sage: (k.ideal(M[1], M[3])*alpha.ideal()).is_principal()
             True
-            sage: M[0] == alpha.numerator() and M[2]==alpha.denominator()
+            sage: M[0] == alpha.numerator() and M[2] == alpha.denominator()
             True
             sage: NFCusp(k, oo).apply(M) == alpha
             True
@@ -1249,7 +1250,8 @@ def units_mod_ideal(I):
         sage: from sage.modular.cusps_nf import units_mod_ideal
         sage: k.<a> = NumberField(x^3 + 11)
         sage: k.unit_group()
-        Unit group with structure C2 x Z of Number Field in a with defining polynomial x^3 + 11
+        Unit group with structure C2 x Z of
+         Number Field in a with defining polynomial x^3 + 11
         sage: I = k.ideal(5, a + 1)
         sage: units_mod_ideal(I)
         [1,
@@ -1261,7 +1263,8 @@ def units_mod_ideal(I):
         sage: from sage.modular.cusps_nf import units_mod_ideal
         sage: k.<a> = NumberField(x^4 - x^3 -21*x^2 + 17*x + 133)
         sage: k.unit_group()
-        Unit group with structure C6 x Z of Number Field in a with defining polynomial x^4 - x^3 - 21*x^2 + 17*x + 133
+        Unit group with structure C6 x Z of
+         Number Field in a with defining polynomial x^4 - x^3 - 21*x^2 + 17*x + 133
         sage: I = k.ideal(3)
         sage: U = units_mod_ideal(I)
         sage: all(U[j].is_unit() and (U[j] not in I) for j in range(len(U)))

src/sage/modular/modform/element.py

@@ -2315,13 +2315,14 @@ def minimal_twist(self, p=None):
             sage: f.twist(chi, level=11) == g
             True
 
-            sage: f = Newforms(575, 2, names='a')[4]    # long time
-            sage: g, chi = f.minimal_twist(5)           # long time
-            sage: g                                     # long time
+            sage: # long time
+            sage: f = Newforms(575, 2, names='a')[4]
+            sage: g, chi = f.minimal_twist(5)
+            sage: g
             q + a*q^2 - a*q^3 - 2*q^4 + (1/2*a + 2)*q^5 + O(q^6)
-            sage: chi                                   # long time
+            sage: chi
             Dirichlet character modulo 5 of conductor 5 mapping 2 |--> 1/2*a
-            sage: f.twist(chi, level=g.level()) == g    # long time
+            sage: f.twist(chi, level=g.level()) == g
             True
         """
         if p is None:

src/sage/modular/modform_hecketriangle/analytic_type.py

@@ -1,3 +1,4 @@
+# sage.doctest: needs sage.graphs
 r"""
 Analytic types of modular forms
 

src/sage/modular/overconvergent/weightspace.py

@@ -1,4 +1,4 @@
-# -*- coding: utf-8 -*-
+# sage.doctest: needs sage.rings.padics
 r"""
 The space of `p`-adic weights
 
@@ -17,7 +17,8 @@
 
     sage: W = pAdicWeightSpace(17)
     sage: W
-    Space of 17-adic weight-characters defined over 17-adic Field with capped relative precision 20
+    Space of 17-adic weight-characters
+     defined over 17-adic Field with capped relative precision 20
     sage: R.<x> = QQ[]
     sage: L = Qp(17).extension(x^2 - 17, names='a'); L.rename('L')
     sage: W.base_extend(L)
@@ -101,7 +102,8 @@ def WeightSpace_constructor(p, base_ring=None):
     EXAMPLES::
 
         sage: pAdicWeightSpace(3) # indirect doctest
-        Space of 3-adic weight-characters defined over 3-adic Field with capped relative precision 20
+        Space of 3-adic weight-characters
+         defined over 3-adic Field with capped relative precision 20
         sage: pAdicWeightSpace(3, QQ)
         Space of 3-adic weight-characters defined over Rational Field
         sage: pAdicWeightSpace(10)
@@ -248,11 +250,13 @@ def base_extend(self, R):
 
             sage: W = pAdicWeightSpace(3, QQ)
             sage: W.base_extend(Qp(3))
-            Space of 3-adic weight-characters defined over 3-adic Field with capped relative precision 20
+            Space of 3-adic weight-characters
+             defined over 3-adic Field with capped relative precision 20
             sage: W.base_extend(IntegerModRing(12))
             Traceback (most recent call last):
             ...
-            TypeError: No coercion map from 'Rational Field' to 'Ring of integers modulo 12' is defined
+            TypeError: No coercion map from 'Rational Field'
+            to 'Ring of integers modulo 12' is defined
         """
         if R.has_coerce_map_from(self.base_ring()):
             return WeightSpace_constructor(self.prime(), R)
@@ -356,7 +360,9 @@ def pAdicEisensteinSeries(self, ring, prec=20):
 
             sage: kappa = pAdicWeightSpace(3)(3, DirichletGroup(3,QQ).0)
             sage: kappa.pAdicEisensteinSeries(QQ[['q']], 20)
-            1 - 9*q + 27*q^2 - 9*q^3 - 117*q^4 + 216*q^5 + 27*q^6 - 450*q^7 + 459*q^8 - 9*q^9 - 648*q^10 + 1080*q^11 - 117*q^12 - 1530*q^13 + 1350*q^14 + 216*q^15 - 1845*q^16 + 2592*q^17 + 27*q^18 - 3258*q^19 + O(q^20)
+            1 - 9*q + 27*q^2 - 9*q^3 - 117*q^4 + 216*q^5 + 27*q^6 - 450*q^7 + 459*q^8
+             - 9*q^9 - 648*q^10 + 1080*q^11 - 117*q^12 - 1530*q^13 + 1350*q^14 + 216*q^15
+             - 1845*q^16 + 2592*q^17 + 27*q^18 - 3258*q^19 + O(q^20)
         """
         if not self.is_even():
             raise ValueError("Eisenstein series not defined for odd weight-characters")
@@ -570,7 +576,8 @@ def chi(self):
 
             sage: kappa = pAdicWeightSpace(29)(13, DirichletGroup(29, Qp(29)).0^14)
             sage: kappa.chi()
-            Dirichlet character modulo 29 of conductor 29 mapping 2 |--> 28 + 28*29 + 28*29^2 + ... + O(29^20)
+            Dirichlet character modulo 29 of conductor 29
+             mapping 2 |--> 28 + 28*29 + 28*29^2 + ... + O(29^20)
         """
         return self._chi
 
@@ -667,7 +674,8 @@ def Lvalue(self):
             sage: pAdicWeightSpace(7)(5, DirichletGroup(7, Qp(7)).0^4).Lvalue()
             0
             sage: pAdicWeightSpace(7)(6, DirichletGroup(7, Qp(7)).0^4).Lvalue()
-            1 + 2*7 + 7^2 + 3*7^3 + 3*7^5 + 4*7^6 + 2*7^7 + 5*7^8 + 2*7^9 + 3*7^10 + 6*7^11 + 2*7^12 + 3*7^13 + 5*7^14 + 6*7^15 + 5*7^16 + 3*7^17 + 6*7^18 + O(7^19)
+            1 + 2*7 + 7^2 + 3*7^3 + 3*7^5 + 4*7^6 + 2*7^7 + 5*7^8 + 2*7^9 + 3*7^10 + 6*7^11
+             + 2*7^12 + 3*7^13 + 5*7^14 + 6*7^15 + 5*7^16 + 3*7^17 + 6*7^18 + O(7^19)
         """
         if self._k > 0:
             return -self._chi.bernoulli(self._k) / self._k

src/sage/modular/pollack_stevens/padic_lseries.py

@@ -1,4 +1,4 @@
-# -*- coding: utf-8 -*-
+# sage.doctest: needs sage.ring.padics
 r"""
 `p`-adic `L`-series attached to overconvergent eigensymbols
 

src/sage/schemes/affine/affine_homset.py

@@ -211,8 +211,8 @@ def points(self, **kwds):
         EXAMPLES: The bug reported at #11526 is fixed::
 
             sage: A2 = AffineSpace(ZZ, 2)
-            sage: F = GF(3)                                                             # optional - sage.rings.finite_rings
-            sage: A2(F).points()                                                        # optional - sage.rings.finite_rings
+            sage: F = GF(3)
+            sage: A2(F).points()
             [(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]
 
         ::
@@ -228,23 +228,23 @@ def points(self, **kwds):
         ::
 
             sage: u = QQ['u'].0
-            sage: K.<v> = NumberField(u^2 + 3)                                          # optional - sage.rings.number_field
-            sage: A.<x,y> = AffineSpace(K, 2)                                           # optional - sage.rings.number_field
-            sage: len(A(K).points(bound=2))                                             # optional - sage.rings.number_field
+            sage: K.<v> = NumberField(u^2 + 3)                                          # needs sage.rings.number_field
+            sage: A.<x,y> = AffineSpace(K, 2)                                           # needs sage.rings.number_field
+            sage: len(A(K).points(bound=2))                                             # needs sage.rings.number_field
             1849
 
         ::
 
             sage: A.<x,y> = AffineSpace(QQ, 2)
             sage: E = A.subscheme([x^2 + y^2 - 1, y^2 - x^3 + x^2 + x - 1])
-            sage: E(A.base_ring()).points()
+            sage: E(A.base_ring()).points()                                             # needs sage.libs.singular
             [(-1, 0), (0, -1), (0, 1), (1, 0)]
 
         ::
 
-            sage: A.<x,y> = AffineSpace(CC, 2)
+            sage: A.<x,y> = AffineSpace(CC, 2)                                          # needs sage.rings.real_mpfr
             sage: E = A.subscheme([y^3 - x^3 - x^2, x*y])
-            sage: E(A.base_ring()).points()
+            sage: E(A.base_ring()).points()                                             # needs sage.libs.singular sage.rings.real_mpfr
             verbose 0 (...: affine_homset.py, points)
             Warning: computations in the numerical fields are inexact;points
             may be computed partially or incorrectly.
@@ -253,9 +253,9 @@ def points(self, **kwds):
 
         ::
 
-            sage: A.<x1,x2> = AffineSpace(CDF, 2)
-            sage: E = A.subscheme([x1^2 + x2^2 + x1*x2, x1 + x2])
-            sage: E(A.base_ring()).points()
+            sage: A.<x1,x2> = AffineSpace(CDF, 2)                                       # needs sage.rings.complex_double
+            sage: E = A.subscheme([x1^2 + x2^2 + x1*x2, x1 + x2])                       # needs sage.libs.singular sage.rings.complex_double
+            sage: E(A.base_ring()).points()                                             # needs sage.libs.singular sage.rings.complex_double
             verbose 0 (...: affine_homset.py, points)
             Warning: computations in the numerical fields are inexact;points
             may be computed partially or incorrectly.
@@ -393,28 +393,29 @@ def numerical_points(self, F=None, **kwds):
 
         EXAMPLES::
 
-            sage: K.<v> = QuadraticField(3)                                             # optional - sage.rings.number_field
-            sage: A.<x,y> = AffineSpace(K, 2)                                           # optional - sage.rings.number_field
-            sage: X = A.subscheme([x^3 - v^2*y, y - v*x^2 + 3])                         # optional - sage.rings.number_field
-            sage: L = X(K).numerical_points(F=RR); L  # abs tol 1e-14                   # optional - sage.rings.number_field
+            sage: # needs sage.rings.number_field
+            sage: K.<v> = QuadraticField(3)
+            sage: A.<x,y> = AffineSpace(K, 2)
+            sage: X = A.subscheme([x^3 - v^2*y, y - v*x^2 + 3])
+            sage: L = X(K).numerical_points(F=RR); L  # abs tol 1e-14
             [(-1.18738247880014, -0.558021142104134),
              (1.57693558184861, 1.30713548084184),
              (4.80659931965815, 37.0162574656220)]
-            sage: L[0].codomain()                                                       # optional - sage.rings.number_field
+            sage: L[0].codomain()
             Affine Space of dimension 2 over Real Field with 53 bits of precision
 
         ::
 
             sage: A.<x,y> = AffineSpace(QQ, 2)
             sage: X = A.subscheme([y^2 - x^2 - 3*x, x^2 - 10*y])
-            sage: len(X(QQ).numerical_points(F=ComplexField(100)))
+            sage: len(X(QQ).numerical_points(F=ComplexField(100)))                      # needs sage.libs.singular
             4
 
         ::
 
             sage: A.<x1, x2> = AffineSpace(QQ, 2)
             sage: E = A.subscheme([30*x1^100 + 1000*x2^2 + 2000*x1*x2 + 1, x1 + x2])
-            sage: len(E(A.base_ring()).numerical_points(F=CDF, zero_tolerance=1e-9))
+            sage: len(E(A.base_ring()).numerical_points(F=CDF, zero_tolerance=1e-9))    # needs sage.libs.singular
             100
 
         TESTS::
@@ -430,7 +431,7 @@ def numerical_points(self, F=None, **kwds):
 
             sage: A.<x,y> = AffineSpace(QQ, 2)
             sage: X = A.subscheme([y^2 - x^2 - 3*x, x^2 - 10*y])
-            sage: X(QQ).numerical_points(F=CC, zero_tolerance=-1)
+            sage: X(QQ).numerical_points(F=CC, zero_tolerance=-1)                       # needs sage.libs.singular
             Traceback (most recent call last):
             ...
             ValueError: tolerance must be positive