path: root/pari-2.17.patch

diff --git a/src/sage/arith/misc.py b/src/sage/arith/misc.py
index 745d5fcbbe7..86409ab3243 100644
--- a/src/sage/arith/misc.py
+++ b/src/sage/arith/misc.py
@@ -2691,9 +2691,14 @@ def factor(n, proof=None, int_=False, algorithm='pari', verbose=0, **kwds):
 
     Any object which has a factor method can be factored like this::
 
-        sage: K.<i> = QuadraticField(-1)                                                # needs sage.rings.number_field
-        sage: factor(122 - 454*i)                                                       # needs sage.rings.number_field
-        (-i) * (-i - 2)^3 * (i + 1)^3 * (-2*i + 3) * (i + 4)
+        sage: # needs sage.rings.number_field
+        sage: K.<i> = QuadraticField(-1)
+        sage: f = factor(122 - 454*i); f                                                # random
+        (i) * (i - 1)^3 * (i + 2)^3 * (3*i + 2) * (i + 4)
+        sage: len(f)
+        4
+        sage: product(p[0]^p[1] for p in f) * f.unit()
+        -454*i + 122
 
     To access the data in a factorization::
 
@@ -2775,8 +2780,10 @@ def radical(n, *args, **kwds):
         ...
         ArithmeticError: radical of 0 is not defined
         sage: K.<i> = QuadraticField(-1)                                                # needs sage.rings.number_field
-        sage: radical(K(2))                                                             # needs sage.rings.number_field
-        i + 1
+        sage: r = radical(K(2)); r                                                      # random, needs sage.rings.number_field
+        i - 1
+        sage: r.norm()                                                                  # needs sage.rings.number_field
+        2
 
     Tests with numpy and gmpy2 numbers::
 
@@ -3031,7 +3038,7 @@ def is_squarefree(n):
         sage: is_squarefree(O(2))
         False
         sage: O(2).factor()
-        (-I) * (I + 1)^2
+        (...) * (...)^2
 
     This method fails on domains which are not Unique Factorization Domains::
 
diff --git a/src/sage/categories/quotient_fields.py b/src/sage/categories/quotient_fields.py
index 76f0570a819..0e4d13ef889 100644
--- a/src/sage/categories/quotient_fields.py
+++ b/src/sage/categories/quotient_fields.py
@@ -100,7 +100,7 @@ class QuotientFields(Category_singleton):
                 sage: R = ZZ.extension(x^2 + 1, names='i')
                 sage: i = R.1
                 sage: gcd(5, 3 + 4*i)
-                -i - 2
+                2*i - 1
                 sage: P.<t> = R[]
                 sage: gcd(t, i)
                 Traceback (most recent call last):
diff --git a/src/sage/dynamics/arithmetic_dynamics/berkovich_ds.py b/src/sage/dynamics/arithmetic_dynamics/berkovich_ds.py
index 6995f61f463..457b0ba2253 100644
--- a/src/sage/dynamics/arithmetic_dynamics/berkovich_ds.py
+++ b/src/sage/dynamics/arithmetic_dynamics/berkovich_ds.py
@@ -690,10 +690,10 @@ class DynamicalSystem_Berkovich_projective(DynamicalSystem_Berkovich):
 
             sage: # needs sage.rings.number_field
             sage: ideal = A.ideal(5).factor()[1][0]; ideal
-            Fractional ideal (2*a + 1)
+            Fractional ideal (-a + 2)
             sage: g = f.conjugate(conj, new_ideal=ideal)
             sage: g.domain().ideal()
-            Fractional ideal (2*a + 1)
+            Fractional ideal (-a + 2)
         """
         if self.domain().is_padic_base():
             return DynamicalSystem_Berkovich(self._system.conjugate(M, adjugate=adjugate))
diff --git a/src/sage/dynamics/arithmetic_dynamics/projective_ds.py b/src/sage/dynamics/arithmetic_dynamics/projective_ds.py
index 262c063a35d..e2c43cc73f8 100644
--- a/src/sage/dynamics/arithmetic_dynamics/projective_ds.py
+++ b/src/sage/dynamics/arithmetic_dynamics/projective_ds.py
@@ -1790,7 +1790,7 @@ class DynamicalSystem_projective(SchemeMorphism_polynomial_projective_space,
             sage: P.<x,y> = ProjectiveSpace(K,1)
             sage: f = DynamicalSystem_projective([1/3*x^2+1/a*y^2, y^2])
             sage: f.primes_of_bad_reduction()                                           # needs sage.rings.function_field
-            [Fractional ideal (a), Fractional ideal (3)]
+            [Fractional ideal (-a), Fractional ideal (3)]
 
         This is an example where ``check=False`` returns extra primes::
 
diff --git a/src/sage/libs/pari/convert_sage.pyx b/src/sage/libs/pari/convert_sage.pyx
index 64386bcf632..db6725f39a5 100644
--- a/src/sage/libs/pari/convert_sage.pyx
+++ b/src/sage/libs/pari/convert_sage.pyx
@@ -573,17 +573,16 @@ cpdef list pari_prime_range(long c_start, long c_stop, bint py_ints=False):
         sage: pari_prime_range(2, 19)
         [2, 3, 5, 7, 11, 13, 17]
     """
-    cdef long p = 0
-    cdef byteptr pari_prime_ptr = diffptr
+    cdef ulong i = 1
     res = []
-    while p < c_start:
-        NEXT_PRIME_VIADIFF(p, pari_prime_ptr)
-    while p < c_stop:
+    while pari_PRIMES[i] < c_start:
+        i+=1
+    while pari_PRIMES[i] < c_stop:
         if py_ints:
-            res.append(p)
+            res.append(pari_PRIMES[i])
         else:
             z = <Integer>PY_NEW(Integer)
-            mpz_set_ui(z.value, p)
+            mpz_set_ui(z.value, pari_PRIMES[i])
             res.append(z)
-        NEXT_PRIME_VIADIFF(p, pari_prime_ptr)
+        i+=1
     return res
diff --git a/src/sage/libs/pari/convert_sage_real_mpfr.pyx b/src/sage/libs/pari/convert_sage_real_mpfr.pyx
index 98db6023dc9..5fd7fba1c47 100644
--- a/src/sage/libs/pari/convert_sage_real_mpfr.pyx
+++ b/src/sage/libs/pari/convert_sage_real_mpfr.pyx
@@ -28,7 +28,7 @@ cpdef Gen new_gen_from_real_mpfr_element(RealNumber self):
 
     # We round up the precision to the nearest multiple of wordsize.
     cdef int rounded_prec
-    rounded_prec = (self.prec() + wordsize - 1) & ~(wordsize - 1)
+    rounded_prec = nbits2prec(self.prec())
 
     # Yes, assigning to self works fine, even in Cython.
     if rounded_prec > prec:
@@ -48,7 +48,7 @@ cpdef Gen new_gen_from_real_mpfr_element(RealNumber self):
         exponent = mpfr_get_z_exp(mantissa, self.value)
 
         # Create a PARI REAL
-        pari_float = cgetr(2 + rounded_prec / wordsize)
+        pari_float = cgetr(rounded_prec)
         pari_float[1] = evalexpo(exponent + rounded_prec - 1) + evalsigne(mpfr_sgn(self.value))
         mpz_export(&pari_float[2], NULL, 1, wordsize // 8, 0, 0, mantissa)
         mpz_clear(mantissa)
diff --git a/src/sage/libs/pari/tests.py b/src/sage/libs/pari/tests.py
index 1ed571cd4b9..bd8dc9641d2 100644
--- a/src/sage/libs/pari/tests.py
+++ b/src/sage/libs/pari/tests.py
@@ -1502,7 +1502,7 @@ Quadratic class numbers::
     sage: pari(-104).quadclassunit()
     [6, [6], [Qfb(5, -4, 6)], 1]
     sage: pari(109).quadclassunit()
-    [1, [], [], 5.56453508676047]
+    [1, [], [], 5.56453508676047, -1]
     sage: pari(10001).quadclassunit() # random generators
     [16, [16], [Qfb(10, 99, -5, 0.E-38)], 5.29834236561059]
     sage: pari(10001).quadclassunit()[0]
@@ -1749,13 +1749,13 @@ General number fields::
     sage: y = QQ['yy'].0; _ = pari(y) # pari has variable ordering rules
     sage: x = QQ['zz'].0; nf = pari(x^2 + 2).nfinit()
     sage: nf.nfroots(y^2 + 2)
-    [Mod(-zz, zz^2 + 2), Mod(zz, zz^2 + 2)]
+    [Mod(-zz, zz^2 + 2), Mod(zz, zz^2 + 2)]~
     sage: nf = pari(x^3 + 2).nfinit()
     sage: nf.nfroots(y^3 + 2)
-    [Mod(zz, zz^3 + 2)]
+    [Mod(zz, zz^3 + 2)]~
     sage: nf = pari(x^4 + 2).nfinit()
     sage: nf.nfroots(y^4 + 2)
-    [Mod(-zz, zz^4 + 2), Mod(zz, zz^4 + 2)]
+    [Mod(-zz, zz^4 + 2), Mod(zz, zz^4 + 2)]~
 
     sage: nf = pari('x^2 + 1').nfinit()
     sage: nf.nfrootsof1()
diff --git a/src/sage/matrix/matrix2.pyx b/src/sage/matrix/matrix2.pyx
index 0c257cfaf33..780cae7fbf2 100644
--- a/src/sage/matrix/matrix2.pyx
+++ b/src/sage/matrix/matrix2.pyx
@@ -16583,7 +16583,7 @@ cdef class Matrix(Matrix1):
             ....:   -2*a^2 + 4*a - 2, -2*a^2 + 1, 2*a, a^2 - 6, 3*a^2 - a ])
             sage: r,s,p = m._echelon_form_PID()
             sage: s[2]
-            (0, 0, -3*a^2 - 18*a + 34, -68*a^2 + 134*a - 53, -111*a^2 + 275*a - 90)
+            (0, 0, 3*a^2 + 18*a - 34, 68*a^2 - 134*a + 53, 111*a^2 - 275*a + 90)
             sage: r * m == s and r.det() == 1
             True
 
diff --git a/src/sage/modular/cusps_nf.py b/src/sage/modular/cusps_nf.py
index 4d120c075da..f3a3ff7e65b 100644
--- a/src/sage/modular/cusps_nf.py
+++ b/src/sage/modular/cusps_nf.py
@@ -1184,9 +1184,9 @@ def NFCusps_ideal_reps_for_levelN(N, nlists=1):
         sage: from sage.modular.cusps_nf import NFCusps_ideal_reps_for_levelN
         sage: NFCusps_ideal_reps_for_levelN(N)
         [(Fractional ideal (1),
-          Fractional ideal (67, a + 17),
-          Fractional ideal (127, a + 48),
-          Fractional ideal (157, a - 19))]
+          Fractional ideal (67, -4/7*a^3 + 13/7*a^2 + 39/7*a - 43),
+          Fractional ideal (127, -4/7*a^3 + 13/7*a^2 + 39/7*a - 42),
+          Fractional ideal (157, -4/7*a^3 + 13/7*a^2 + 39/7*a + 48))]
         sage: L = NFCusps_ideal_reps_for_levelN(N, 5)
         sage: all(len(L[i]) == k.class_number() for i in range(len(L)))
         True
@@ -1244,7 +1244,7 @@ def units_mod_ideal(I):
         sage: I = k.ideal(5, a + 1)
         sage: units_mod_ideal(I)
         [1,
-        -2*a^2 - 4*a + 1,
+        2*a^2 + 4*a - 1,
         ...]
 
     ::
diff --git a/src/sage/modular/dirichlet.py b/src/sage/modular/dirichlet.py
index 1f6a7a94444..1cf3ccdb611 100644
--- a/src/sage/modular/dirichlet.py
+++ b/src/sage/modular/dirichlet.py
@@ -2395,13 +2395,13 @@ class DirichletGroupFactory(UniqueFactory):
         sage: parent(val)
         Gaussian Integers generated by zeta4 in Cyclotomic Field of order 4 and degree 2
         sage: r4_29_0 = r4.residue_field(K(29).factor()[0][0]); r4_29_0(val)
-        17
+        12
         sage: r4_29_0(val) * GF(29)(3)
-        22
+        7
         sage: r4_29_0(G.gens()[2].values_on_gens()[2]) * 3
-        22
+        7
         sage: parent(r4_29_0(G.gens()[2].values_on_gens()[2]) * 3)
-        Residue field of Fractional ideal (-2*zeta4 + 5)
+        Residue field of Fractional ideal (-2*zeta4 - 5)
 
     ::
 
diff --git a/src/sage/modular/modsym/p1list_nf.py b/src/sage/modular/modsym/p1list_nf.py
index 86d33071974..00bb0979ea4 100644
--- a/src/sage/modular/modsym/p1list_nf.py
+++ b/src/sage/modular/modsym/p1list_nf.py
@@ -61,7 +61,7 @@ Lift an MSymbol to a matrix in `SL(2, R)`:
 
     sage: alpha = MSymbol(N, a + 2, 3*a^2)
     sage: alpha.lift_to_sl2_Ok()
-    [-1, 4*a^2 - 13*a + 23, a + 2, 5*a^2 + 3*a - 3]
+    [-a - 1, 15*a^2 - 38*a + 86, a + 2, -a^2 + 9*a - 19]
     sage: Ok = k.ring_of_integers()
     sage: M = Matrix(Ok, 2, alpha.lift_to_sl2_Ok())
     sage: det(M)
@@ -977,11 +977,11 @@ class P1NFList(SageObject):
             sage: N = k.ideal(5, a + 1)
             sage: P = P1NFList(N)
             sage: u = k.unit_group().gens_values(); u
-            [-1, -2*a^2 - 4*a + 1]
+            [-1, 2*a^2 + 4*a - 1]
             sage: P.apply_J_epsilon(4, -1)
             2
             sage: P.apply_J_epsilon(4, u[0], u[1])
-            5
+            1
 
         ::
 
@@ -1122,7 +1122,7 @@ def lift_to_sl2_Ok(N, c, d):
         sage: M = Matrix(Ok, 2, lift_to_sl2_Ok(N, 0, 7))
         Traceback (most recent call last):
         ...
-        ValueError: <0> + <7> and the Fractional ideal (7, a) are not coprime.
+        ValueError: <0> + <7> and the Fractional ideal (7, -4/7*a^3 + 13/7*a^2 + 39/7*a - 19) are not coprime.
     """
     k = N.number_field()
     # check the input
diff --git a/src/sage/quadratic_forms/binary_qf.py b/src/sage/quadratic_forms/binary_qf.py
index 1a75f415b64..c89759288b9 100755
--- a/src/sage/quadratic_forms/binary_qf.py
+++ b/src/sage/quadratic_forms/binary_qf.py
@@ -1646,7 +1646,7 @@ class BinaryQF(SageObject):
             sage: Q = BinaryQF([1, 0, 12345])
             sage: n = 2^99 + 5273
             sage: Q.solve_integer(n)                                                    # needs sage.libs.pari
-            (-67446480057659, 7139620553488)
+            (67446480057659, 7139620553488)
             sage: Q.solve_integer(n, algorithm='cornacchia')                            # needs sage.libs.pari
             (67446480057659, 7139620553488)
             sage: timeit('Q.solve_integer(n)')                          # not tested
@@ -1661,7 +1661,7 @@ class BinaryQF(SageObject):
             sage: Qs
             [x^2 + x*y + 6*y^2, 2*x^2 - x*y + 3*y^2, 2*x^2 + x*y + 3*y^2]
             sage: [Q.solve_integer(3) for Q in Qs]
-            [None, (0, -1), (0, -1)]
+            [None, (0, 1), (0, 1)]
             sage: [Q.solve_integer(5) for Q in Qs]
             [None, None, None]
             sage: [Q.solve_integer(6) for Q in Qs]
@@ -1741,11 +1741,11 @@ class BinaryQF(SageObject):
             sage: # needs sage.libs.pari
             sage: Q = BinaryQF([1, 0, 5])
             sage: Q.solve_integer(126, _flag=1)
-            [(11, -1), (-1, -5), (-1, 5), (-11, -1)]
+            [(-11, -1), (-1, -5), (-1, 5), (11, -1)]
             sage: Q.solve_integer(126, _flag=2)
             (11, -1)
             sage: Q.solve_integer(126, _flag=3)
-            [(11, -1), (-1, -5), (-1, 5), (-11, -1), (-9, -3), (9, -3)]
+            [(-11, -1), (-9, -3), (-1, -5), (-1, 5), (9, -3), (11, -1)]
         """
         if self.is_negative_definite():  # not supported by PARI
             return (-self).solve_integer(-n)
diff --git a/src/sage/rings/finite_rings/finite_field_prime_modn.py b/src/sage/rings/finite_rings/finite_field_prime_modn.py
index d94b0a4335a..0978c7328fe 100644
--- a/src/sage/rings/finite_rings/finite_field_prime_modn.py
+++ b/src/sage/rings/finite_rings/finite_field_prime_modn.py
@@ -114,9 +114,9 @@ class FiniteField_prime_modn(FiniteField_generic, integer_mod_ring.IntegerModRin
             sage: RF13 = K.residue_field(pp)
             sage: RF13.hom([GF(13)(1)])
             Ring morphism:
-             From: Residue field of Fractional ideal (-w - 18)
-             To:   Finite Field of size 13
-             Defn: 1 |--> 1
+              From: Residue field of Fractional ideal (w + 18)
+              To:   Finite Field of size 13
+              Defn: 1 |--> 1
 
         Check that :issue:`19573` is resolved::
 
diff --git a/src/sage/rings/finite_rings/residue_field.pyx b/src/sage/rings/finite_rings/residue_field.pyx
index 3146f7fd764..a8e77e48b76 100644
--- a/src/sage/rings/finite_rings/residue_field.pyx
+++ b/src/sage/rings/finite_rings/residue_field.pyx
@@ -22,14 +22,13 @@ monogenic (i.e., 2 is an essential discriminant divisor)::
     sage: # needs sage.rings.number_field
     sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8)
     sage: F = K.factor(2); F
-    (Fractional ideal (-1/2*a^2 + 1/2*a - 1)) * (Fractional ideal (-a^2 + 2*a - 3))
-    * (Fractional ideal (3/2*a^2 - 5/2*a + 4))
+    (Fractional ideal (-1/2*a^2 + 1/2*a - 1)) * (Fractional ideal (a^2 - 2*a + 3)) * (Fractional ideal (-3/2*a^2 + 5/2*a - 4))
     sage: F[0][0].residue_field()
     Residue field of Fractional ideal (-1/2*a^2 + 1/2*a - 1)
     sage: F[1][0].residue_field()
-    Residue field of Fractional ideal (-a^2 + 2*a - 3)
+    Residue field of Fractional ideal (a^2 - 2*a + 3)
     sage: F[2][0].residue_field()
-    Residue field of Fractional ideal (3/2*a^2 - 5/2*a + 4)
+    Residue field of Fractional ideal (-3/2*a^2 + 5/2*a - 4)
 
 We can also form residue fields from `\ZZ`::
 
@@ -126,10 +125,10 @@ First over a small non-prime field::
     sage: I = ideal([ubar*X + Y]); I
     Ideal (ubar*X + Y) of Multivariate Polynomial Ring in X, Y over
      Residue field in ubar of Fractional ideal
-      (47, 517/55860*u^5 + 235/3724*u^4 + 9829/13965*u^3
-            + 54106/13965*u^2 + 64517/27930*u + 755696/13965)
+      (47, 4841/93100*u^5 + 34451/139650*u^4 + 303697/69825*u^3
+            + 297893/27930*u^2 + 1649764/23275*u + 2633506/69825)
     sage: I.groebner_basis()                                                            # needs sage.libs.singular
-    [X + (-19*ubar^2 - 5*ubar - 17)*Y]
+    [X + (-15*ubar^2 + 3*ubar - 2)*Y]
 
 And now over a large prime field::
 
@@ -496,9 +495,9 @@ class ResidueField_generic(Field):
 
         sage: # needs sage.rings.number_field
         sage: I = QQ[i].factor(2)[0][0]; I
-        Fractional ideal (I + 1)
+        Fractional ideal (-I - 1)
         sage: k = I.residue_field(); k
-        Residue field of Fractional ideal (I + 1)
+        Residue field of Fractional ideal (-I - 1)
         sage: type(k)
         <class 'sage.rings.finite_rings.residue_field.ResidueFiniteField_prime_modn_with_category'>
 
@@ -1008,7 +1007,7 @@ cdef class ReductionMap(Map):
             sage: cr
             Partially defined reduction map:
               From: Number Field in a with defining polynomial x^2 + 1
-              To:   Residue field of Fractional ideal (a + 1)
+              To:   Residue field of Fractional ideal (-a + 1)
             sage: cr == r                       # not implemented
             True
             sage: r(2 + a) == cr(2 + a)
@@ -1039,7 +1038,7 @@ cdef class ReductionMap(Map):
             sage: cr
             Partially defined reduction map:
               From: Number Field in a with defining polynomial x^2 + 1
-              To:   Residue field of Fractional ideal (a + 1)
+              To:   Residue field of Fractional ideal (-a + 1)
             sage: cr == r                       # not implemented
             True
             sage: r(2 + a) == cr(2 + a)
@@ -1071,7 +1070,7 @@ cdef class ReductionMap(Map):
             sage: r = F.reduction_map(); r
             Partially defined reduction map:
               From: Number Field in a with defining polynomial x^2 + 1
-              To:   Residue field of Fractional ideal (a + 1)
+              To:   Residue field of Fractional ideal (-a + 1)
 
         We test that calling the function also works after copying::
 
@@ -1083,7 +1082,7 @@ cdef class ReductionMap(Map):
             Traceback (most recent call last):
             ...
             ZeroDivisionError: Cannot reduce field element 1/2*a
-            modulo Fractional ideal (a + 1): it has negative valuation
+            modulo Fractional ideal (-a + 1): it has negative valuation
 
             sage: # needs sage.rings.finite_rings
             sage: R.<t> = GF(2)[]; h = t^5 + t^2 + 1
@@ -1105,11 +1104,11 @@ cdef class ReductionMap(Map):
             sage: # needs sage.rings.number_field
             sage: K.<i> = NumberField(x^2 + 1)
             sage: P1, P2 = [g[0] for g in K.factor(5)]; P1, P2
-            (Fractional ideal (-i - 2), Fractional ideal (2*i + 1))
+            (Fractional ideal (2*i - 1), Fractional ideal (-2*i - 1))
             sage: a = 1/(1+2*i)
             sage: F1, F2 = [g.residue_field() for g in [P1,P2]]; F1, F2
-            (Residue field of Fractional ideal (-i - 2),
-             Residue field of Fractional ideal (2*i + 1))
+            (Residue field of Fractional ideal (2*i - 1),
+             Residue field of Fractional ideal (-2*i - 1))
             sage: a.valuation(P1)
             0
             sage: F1(i/7)
@@ -1122,7 +1121,7 @@ cdef class ReductionMap(Map):
             Traceback (most recent call last):
             ...
             ZeroDivisionError: Cannot reduce field element -2/5*i + 1/5
-            modulo Fractional ideal (2*i + 1): it has negative valuation
+            modulo Fractional ideal (-2*i - 1): it has negative valuation
         """
         # The reduction map is just x |--> F(to_vs(x) * (PB**(-1))) if
         # either x is integral or the denominator of x is coprime to
@@ -1188,8 +1187,7 @@ cdef class ReductionMap(Map):
             sage: f = k.convert_map_from(K)
             sage: s = f.section(); s
             Lifting map:
-              From: Residue field in abar of
-                    Fractional ideal (-14*a^4 + 24*a^3 + 26*a^2 - 58*a + 15)
+              From: Residue field in abar of Fractional ideal (14*a^4 - 24*a^3 - 26*a^2 + 58*a - 15)
               To:   Number Field in a with defining polynomial x^5 - 5*x + 2
             sage: s(k.gen())
             a
@@ -1424,8 +1422,7 @@ cdef class ResidueFieldHomomorphism_global(RingHomomorphism):
             sage: f = k.coerce_map_from(K.ring_of_integers())
             sage: s = f.section(); s
             Lifting map:
-              From: Residue field in abar of
-                    Fractional ideal (-14*a^4 + 24*a^3 + 26*a^2 - 58*a + 15)
+              From: Residue field in abar of Fractional ideal (14*a^4 - 24*a^3 - 26*a^2 + 58*a - 15)
               To:   Maximal Order generated by a in Number Field in a with defining polynomial x^5 - 5*x + 2
             sage: s(k.gen())
             a
@@ -1678,7 +1675,7 @@ cdef class LiftingMap(Section):
             sage: F.<tmod> = K.factor(7)[0][0].residue_field()
             sage: F.lift_map() #indirect doctest
             Lifting map:
-              From: Residue field in tmod of Fractional ideal (theta_12^2 + 2)
+              From: Residue field in tmod of Fractional ideal (2*theta_12^3 + theta_12)
               To:   Maximal Order generated by theta_12 in Cyclotomic Field of order 12 and degree 4
         """
         return "Lifting"
diff --git a/src/sage/rings/finite_rings/residue_field_pari_ffelt.pyx b/src/sage/rings/finite_rings/residue_field_pari_ffelt.pyx
index e9962c3ccde..90a68c619f6 100644
--- a/src/sage/rings/finite_rings/residue_field_pari_ffelt.pyx
+++ b/src/sage/rings/finite_rings/residue_field_pari_ffelt.pyx
@@ -103,7 +103,7 @@ class ResidueFiniteField_pari_ffelt(ResidueField_generic, FiniteField_pari_ffelt
             sage: P.residue_class_degree()
             2
             sage: ff.<alpha> = P.residue_field(); ff
-            Residue field in alpha of Fractional ideal (-12*aa^2 + 189*aa - 475)
+            Residue field in alpha of Fractional ideal (12*aa^2 - 189*aa + 475)
             sage: type(ff)
             <class 'sage.rings.finite_rings.residue_field_pari_ffelt.ResidueFiniteField_pari_ffelt_with_category'>
             sage: ff(alpha^2 + 1)
diff --git a/src/sage/rings/integer.pyx b/src/sage/rings/integer.pyx
index 9616e7946bc..bcc45e703a2 100644
--- a/src/sage/rings/integer.pyx
+++ b/src/sage/rings/integer.pyx
@@ -5585,7 +5585,7 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
             sage: 5.is_norm(K)
             False
             sage: n.is_norm(K, element=True)
-            (True, -4*beta + 6)
+            (True, 4*beta + 6)
             sage: n.is_norm(K, element=True)[1].norm()
             4
             sage: n = 5
diff --git a/src/sage/rings/number_field/S_unit_solver.py b/src/sage/rings/number_field/S_unit_solver.py
index 0ffac369720..1260650ef77 100644
--- a/src/sage/rings/number_field/S_unit_solver.py
+++ b/src/sage/rings/number_field/S_unit_solver.py
@@ -12,10 +12,10 @@ EXAMPLES::
     sage: x = polygen(ZZ, 'x')
     sage: K.<xi> = NumberField(x^2 + x + 1)
     sage: S = K.primes_above(3)
-    sage: expected = [((0, 1), (4, 0), xi + 2, -xi - 1),
-    ....:             ((1, -1), (0, -1), 1/3*xi + 2/3, -1/3*xi + 1/3),
+    sage: expected = [((4, 1), (4, 0), xi + 2, -xi - 1),
+    ....:             ((3, -1), (2, -1), 1/3*xi + 2/3, -1/3*xi + 1/3),
     ....:             ((1, 0), (5, 0), xi + 1, -xi),
-    ....:             ((2, 0), (5, 1), xi, -xi + 1)]
+    ....:             ((2, 0), (3, 1), xi, -xi + 1)]
     sage: sols = solve_S_unit_equation(K, S, 200)
     sage: eq_up_to_order(sols, expected)
     True
@@ -1448,7 +1448,7 @@ def embedding_to_Kp(a, prime, prec):
         sage: from sage.rings.number_field.S_unit_solver import embedding_to_Kp
         sage: K.<a> = QuadraticField(17)
         sage: p = K.prime_above(13); p
-        Fractional ideal (-a + 2)
+        Fractional ideal (a - 2)
         sage: embedding_to_Kp(a-3, p, 15)
         -20542890112375827
 
@@ -1791,10 +1791,10 @@ def sieve_ordering(SUK, q):
          Residue field of Fractional ideal (2*xi + 1))
 
         sage: sieve_data[2]
-        ([18, 12, 16, 8], [18, 16, 10, 4], [18, 10, 12, 10])
+        ([18, 9, 16, 8], [18, 7, 10, 4], [18, 3, 12, 10])
 
         sage: sieve_data[3]
-        (648, 2916, 3888)
+        (972, 972, 3888)
     """
 
     K = SUK.number_field()
@@ -2170,23 +2170,23 @@ def construct_complement_dictionaries(split_primes_list, SUK, verbose=False):
         sage: SUK = K.S_unit_group(S=K.primes_above(H))
         sage: split_primes_list = [3, 7]
         sage: actual = construct_complement_dictionaries(split_primes_list, SUK)
-        sage: expected = {3: {(0, 1, 0): [(1, 0, 0), (0, 1, 0)],
-        ....:                 (1, 0, 0): [(1, 0, 0), (0, 1, 0)]},
-        ....:             7: {(0, 1, 0): [(1, 0, 0), (1, 4, 4), (1, 2, 2)],
-        ....:                 (0, 1, 2): [(0, 1, 2), (0, 3, 4), (0, 5, 0)],
-        ....:                 (0, 3, 2): [(1, 0, 0), (1, 4, 4), (1, 2, 2)],
-        ....:                 (0, 3, 4): [(0, 1, 2), (0, 3, 4), (0, 5, 0)],
-        ....:                 (0, 5, 0): [(0, 1, 2), (0, 3, 4), (0, 5, 0)],
-        ....:                 (0, 5, 4): [(1, 0, 0), (1, 4, 4), (1, 2, 2)],
-        ....:                 (1, 0, 0): [(0, 5, 4), (0, 3, 2), (0, 1, 0)],
-        ....:                 (1, 0, 2): [(1, 0, 4), (1, 4, 2), (1, 2, 0)],
+        sage: expected = {3: {(1, 1, 0): [(1, 0, 0), (1, 1, 0)],
+        ....:                 (1, 0, 0): [(1, 0, 0), (1, 1, 0)]},
+        ....:             7: {(1, 1, 0): [(1, 0, 0), (1, 4, 4), (1, 2, 2)],
+        ....:                 (1, 3, 2): [(1, 0, 0), (1, 4, 4), (1, 2, 2)],
+        ....:                 (1, 5, 4): [(1, 0, 0), (1, 4, 4), (1, 2, 2)],
         ....:                 (1, 0, 4): [(1, 2, 4), (1, 4, 0), (1, 0, 2)],
         ....:                 (1, 2, 0): [(1, 2, 4), (1, 4, 0), (1, 0, 2)],
-        ....:                 (1, 2, 2): [(0, 5, 4), (0, 3, 2), (0, 1, 0)],
+        ....:                 (1, 4, 2): [(1, 2, 4), (1, 4, 0), (1, 0, 2)],
+        ....:                 (1, 1, 2): [(1, 1, 2), (1, 3, 4), (1, 5, 0)],
+        ....:                 (1, 3, 4): [(1, 1, 2), (1, 3, 4), (1, 5, 0)],
+        ....:                 (1, 5, 0): [(1, 1, 2), (1, 3, 4), (1, 5, 0)],
+        ....:                 (1, 0, 2): [(1, 0, 4), (1, 4, 2), (1, 2, 0)],
         ....:                 (1, 2, 4): [(1, 0, 4), (1, 4, 2), (1, 2, 0)],
         ....:                 (1, 4, 0): [(1, 0, 4), (1, 4, 2), (1, 2, 0)],
-        ....:                 (1, 4, 2): [(1, 2, 4), (1, 4, 0), (1, 0, 2)],
-        ....:                 (1, 4, 4): [(0, 5, 4), (0, 3, 2), (0, 1, 0)]}}
+        ....:                 (1, 0, 0): [(1, 5, 4), (1, 3, 2), (1, 1, 0)],
+        ....:                 (1, 2, 2): [(1, 5, 4), (1, 3, 2), (1, 1, 0)],
+        ....:                 (1, 4, 4): [(1, 5, 4), (1, 3, 2), (1, 1, 0)]}}
         sage: all(set(actual[p][vec]) == set(expected[p][vec])
         ....:     for p in [3, 7] for vec in expected[p])
         True
@@ -2693,9 +2693,9 @@ def sieve_below_bound(K, S, bound=10, bump=10, split_primes_list=[], verbose=Fal
         sage: SUK = UnitGroup(K, S=tuple(K.primes_above(3)))
         sage: S = SUK.primes()
         sage: sols = sieve_below_bound(K, S, 10)
-        sage: expected = [((1, -1), (0, -1), 1/3*xi + 2/3, -1/3*xi + 1/3),
-        ....:             ((0, 1), (4, 0), xi + 2, -xi - 1),
-        ....:             ((2, 0), (5, 1), xi, -xi + 1),
+        sage: expected = [((3, -1), (2, -1), 1/3*xi + 2/3, -1/3*xi + 1/3),
+        ....:             ((4, 1), (4, 0), xi + 2, -xi - 1),
+        ....:             ((2, 0), (3, 1), xi, -xi + 1),
         ....:             ((1, 0), (5, 0), xi + 1, -xi)]
         sage: eq_up_to_order(sols, expected)
         True
@@ -2758,10 +2758,10 @@ def solve_S_unit_equation(K, S, prec=106, include_exponents=True, include_bound=
         sage: K.<xi> = NumberField(x^2 + x + 1)
         sage: S = K.primes_above(3)
         sage: sols = solve_S_unit_equation(K, S, 200)
-        sage: expected = [((0, 1), (4, 0), xi + 2, -xi - 1),
-        ....:             ((1, -1), (0, -1), 1/3*xi + 2/3, -1/3*xi + 1/3),
+        sage: expected = [((4, 1), (4, 0), xi + 2, -xi - 1),
+        ....:             ((3, -1), (2, -1), 1/3*xi + 2/3, -1/3*xi + 1/3),
         ....:             ((1, 0), (5, 0), xi + 1, -xi),
-        ....:             ((2, 0), (5, 1), xi, -xi + 1)]
+        ....:             ((2, 0), (3, 1), xi, -xi + 1)]
         sage: eq_up_to_order(sols, expected)
         True
 
@@ -2769,7 +2769,7 @@ def solve_S_unit_equation(K, S, prec=106, include_exponents=True, include_bound=
 
         sage: solutions, bound = solve_S_unit_equation(K, S, 100, include_bound=True)
         sage: bound
-        7
+        6
 
     You can omit the exponent vectors::
 
diff --git a/src/sage/rings/number_field/galois_group.py b/src/sage/rings/number_field/galois_group.py
index c974c3df6ff..ee53ca6d674 100644
--- a/src/sage/rings/number_field/galois_group.py
+++ b/src/sage/rings/number_field/galois_group.py
@@ -995,8 +995,8 @@ class GaloisGroup_v2(GaloisGroup_perm):
             sage: K.<b> = NumberField(x^4 - 2*x^2 + 2, 'a').galois_closure()
             sage: G = K.galois_group()
             sage: [G.artin_symbol(P) for P in K.primes_above(7)]
-            [(1,4)(2,3)(5,8)(6,7), (1,4)(2,3)(5,8)(6,7),
-             (1,5)(2,6)(3,7)(4,8), (1,5)(2,6)(3,7)(4,8)]
+            [(1,5)(2,6)(3,7)(4,8), (1,5)(2,6)(3,7)(4,8),
+             (1,4)(2,3)(5,8)(6,7), (1,4)(2,3)(5,8)(6,7)]
             sage: G.artin_symbol(17)
             Traceback (most recent call last):
             ...
diff --git a/src/sage/rings/number_field/number_field.py b/src/sage/rings/number_field/number_field.py
index bb16476980e..3a98a997962 100644
--- a/src/sage/rings/number_field/number_field.py
+++ b/src/sage/rings/number_field/number_field.py
@@ -3627,7 +3627,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
             sage: L.<b> = K.extension(x^2 - 3, x^2 + 1)
             sage: M.<c> = L.extension(x^2 + 1)
             sage: L.ideal(K.ideal(2, a))
-            Fractional ideal (a)
+            Fractional ideal (-a)
             sage: M.ideal(K.ideal(2, a)) == M.ideal(a*(b - c)/2)
             True
 
@@ -3665,40 +3665,29 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
 
             sage: x = polygen(QQ, 'x')
             sage: K.<a> = NumberField(x^2 + 23)
-            sage: d = K.ideals_of_bdd_norm(10)
-            sage: for n in d:
-            ....:     print(n)
-            ....:     for I in sorted(d[n]):
-            ....:         print(I)
-            1
-            Fractional ideal (1)
-            2
-            Fractional ideal (2, 1/2*a - 1/2)
-            Fractional ideal (2, 1/2*a + 1/2)
-            3
-            Fractional ideal (3, 1/2*a - 1/2)
-            Fractional ideal (3, 1/2*a + 1/2)
-            4
-            Fractional ideal (2)
-            Fractional ideal (4, 1/2*a + 3/2)
-            Fractional ideal (4, 1/2*a + 5/2)
-            5
-            6
-            Fractional ideal (1/2*a - 1/2)
-            Fractional ideal (1/2*a + 1/2)
-            Fractional ideal (6, 1/2*a + 5/2)
-            Fractional ideal (6, 1/2*a + 7/2)
-            7
-            8
-            Fractional ideal (4, a - 1)
-            Fractional ideal (4, a + 1)
-            Fractional ideal (1/2*a + 3/2)
-            Fractional ideal (1/2*a - 3/2)
-            9
-            Fractional ideal (3)
-            Fractional ideal (9, 1/2*a + 7/2)
-            Fractional ideal (9, 1/2*a + 11/2)
-            10
+            sage: d = K.ideals_of_bdd_norm(10); d # random
+                {1: [Fractional ideal (1)],
+                2: [Fractional ideal (2, 1/2*a - 1/2), Fractional ideal (2, 1/2*a + 1/2)],
+                3: [Fractional ideal (3, 1/2*a + 1/2), Fractional ideal (3, 1/2*a - 1/2)],
+                4: [Fractional ideal (4, 1/2*a + 3/2),
+                 Fractional ideal (2),
+                 Fractional ideal (4, 1/2*a + 5/2)],
+                5: [],
+                6: [Fractional ideal (6, 1/2*a + 7/2),
+                 Fractional ideal (-1/2*a - 1/2),
+                 Fractional ideal (-1/2*a + 1/2),
+                 Fractional ideal (6, 1/2*a + 5/2)],
+                7: [],
+                8: [Fractional ideal (1/2*a + 3/2),
+                 Fractional ideal (4, a - 1),
+                 Fractional ideal (4, a + 1),
+                 Fractional ideal (-1/2*a + 3/2)],
+                9: [Fractional ideal (9, 1/2*a + 7/2),
+                 Fractional ideal (3),
+                 Fractional ideal (9, 1/2*a + 11/2)],
+                10: []}
+            sage: [[I.norm() for I in sorted(d[n])] for n in d]
+                [[1], [2, 2], [3, 3], [4, 4, 4], [], [6, 6, 6, 6], [], [8, 8, 8, 8], [9, 9, 9], []]
         """
         hnf_ideals = self.pari_nf().ideallist(bound)
         d = {}
@@ -3925,9 +3914,13 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
         EXAMPLES::
 
             sage: K.<i> = QuadraticField(-1)
-            sage: K.primes_of_bounded_norm(10)
-            [Fractional ideal (i + 1), Fractional ideal (-i - 2),
-             Fractional ideal (2*i + 1), Fractional ideal (3)]
+            sage: P = K.primes_of_bounded_norm(10); P # random
+            [Fractional ideal (i + 1),
+             Fractional ideal (i + 2),
+             Fractional ideal (-i + 2),
+             Fractional ideal (3)]
+            sage: [p.norm() for p in P]
+            [2, 5, 5, 9]
             sage: K.primes_of_bounded_norm(1)
             []
             sage: x = polygen(QQ, 'x')
@@ -3936,10 +3929,10 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
             sage: P
             [Fractional ideal (a),
              Fractional ideal (a + 1),
-             Fractional ideal (-a^2 - 1),
+             Fractional ideal (a^2 + 1),
              Fractional ideal (a^2 + a - 1),
              Fractional ideal (2*a + 1),
-             Fractional ideal (-2*a^2 - a - 1),
+             Fractional ideal (2*a^2 + a + 1),
              Fractional ideal (a^2 - 2*a - 1),
              Fractional ideal (a + 3)]
             sage: [p.norm() for p in P]
@@ -3988,11 +3981,13 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
 
             sage: K.<i> = QuadraticField(-1)
             sage: it = K.primes_of_bounded_norm_iter(10)
-            sage: list(it)
+            sage: l = list(it); l # random
             [Fractional ideal (i + 1),
              Fractional ideal (3),
-             Fractional ideal (-i - 2),
-             Fractional ideal (2*i + 1)]
+             Fractional ideal (i + 2),
+             Fractional ideal (-i + 2)]
+            sage: [I.norm() for I in l]
+            [2, 9, 5, 5]
             sage: list(K.primes_of_bounded_norm_iter(1))
             []
         """
@@ -4317,7 +4312,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
             sage: k.<a> = NumberField(x^4 - 3/2*x + 5/3); k
             Number Field in a with defining polynomial x^4 - 3/2*x + 5/3
             sage: k.pari_nf()
-            [y^4 - 324*y + 2160, [0, 2], 48918708, 216, ..., [36, 36*y, y^3 + 6*y^2 - 252, 6*y^2], [1, 0, 0, 252; 0, 1, 0, 0; 0, 0, 0, 36; 0, 0, 6, -36], [1, 0, 0, 0, 0, 0, -18, 42, 0, -18, -46, -60, 0, 42, -60, -60; 0, 1, 0, 0, 1, 0, 2, 0, 0, 2, -11, -1, 0, 0, -1, 9; 0, 0, 1, 0, 0, 0, 6, 6, 1, 6, -5, 0, 0, 6, 0, 0; 0, 0, 0, 1, 0, 6, -6, -6, 0, -6, -1, 2, 1, -6, 2, 0]]
+            [y^4 - 324*y + 2160, [0, 2], 48918708, 216, ..., [36, 36*y, y^3 + 6*y^2 - 252, -6*y^2], [1, 0, 0, 252; 0, 1, 0, 0; 0, 0, 0, 36; 0, 0, -6, 36], [1, 0, 0, 0, 0, 0, -18, -42, 0, -18, -46, 60, 0, -42, 60, -60; 0, 1, 0, 0, 1, 0, 2, 0, 0, 2, -11, 1, 0, 0, 1, 9; 0, 0, 1, 0, 0, 0, 6, -6, 1, 6, -5, 0, 0, -6, 0, 0; 0, 0, 0, 1, 0, -6, 6, -6, 0, 6, 1, 2, 1, -6, 2, 0]]
             sage: pari(k)
             [y^4 - 324*y + 2160, [0, 2], 48918708, 216, ...]
             sage: gp(k)
@@ -4807,7 +4802,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
              1/13*a^2 + 7/13*a - 332/13,
              -1/13*a^2 + 6/13*a + 345/13,
              -1,
-             -2/13*a^2 - 1/13*a + 755/13]
+             1/13*a^2 - 19/13*a - 7/13]
             sage: units[5] in (1/13*a^2 - 19/13*a - 7/13, 1/13*a^2 + 20/13*a - 7/13)
             True
             sage: len(units) == 6
@@ -4818,7 +4813,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
 
             sage: K.<a> = NumberField(2*x^2 - 1/3)
             sage: K._S_class_group_and_units(tuple(K.primes_above(2) + K.primes_above(3)))
-            ([6*a + 2, 6*a + 3, -1, -12*a + 5], [])
+            ([...6*a...2, ...6*a...3, -1, ...12*a...5], [])
         """
         K_pari = self.pari_bnf(proof=proof)
         S_pari = [p.pari_prime() for p in sorted(set(S))]
@@ -4996,7 +4991,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
              1/13*a^2 + 7/13*a - 332/13,
              -1/13*a^2 + 6/13*a + 345/13,
              -1,
-             -2/13*a^2 - 1/13*a + 755/13]
+             1/13*a^2 - 19/13*a - 7/13]
             sage: gens[5] in (1/13*a^2 - 19/13*a - 7/13, 1/13*a^2 + 20/13*a - 7/13)
             True
             sage: gens[6] in (-1/13*a^2 + 45/13*a - 97/13, 1/13*a^2 - 45/13*a + 97/13)
@@ -5153,16 +5148,20 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
             sage: KS2, gens, fromKS2, toKS2 = K.selmer_space([P2, P3, P5], 2)
             sage: KS2
             Vector space of dimension 4 over Finite Field of size 2
-            sage: gens
-            [a + 1, a, 2, -1]
+            sage: gens  # random
+            [-a - 1, a, 2, -1]
+            sage: gens in ([-a - 1, a, 2, -1], [a + 1, a, 2, -1])
+            True
 
         Each generator must have even valuation at primes not in `S`::
 
-            sage: [K.ideal(g).factor() for g in gens]
+            sage: facgens = [K.ideal(g).factor() for g in gens]; facgens # random
             [(Fractional ideal (2, a + 1)) * (Fractional ideal (3, a + 1)),
              Fractional ideal (a),
              (Fractional ideal (2, a + 1))^2,
              1]
+            sage: facgens[2][0][1]
+            2
 
             sage: toKS2(10)
             (0, 0, 1, 1)
@@ -5174,8 +5173,10 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
             sage: KS3, gens, fromKS3, toKS3 = K.selmer_space([P2, P3, P5], 3)
             sage: KS3
             Vector space of dimension 3 over Finite Field of size 3
-            sage: gens
-            [1/2, 1/4*a + 1/4, a]
+            sage: gens # random
+            [1/2, -1/4*a - 1/4, a]
+            sage: gens in ([1/2, -1/4*a - 1/4, a], [1/2, 1/4*a + 1/4, a])
+            True
 
         An example to show that the group `K(S,2)` may be strictly
         larger than the group of elements giving extensions unramified
@@ -5640,7 +5641,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
             sage: k.<a> = NumberField(x^2 + 23)
             sage: d = k.different()
             sage: d
-            Fractional ideal (-a)
+            Fractional ideal (a)
             sage: d.norm()
             23
             sage: k.disc()
@@ -5760,7 +5761,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
             sage: K.elements_of_norm(3)
             []
             sage: K.elements_of_norm(50)
-            [-a - 7, 5*a - 5, 7*a + 1]
+            [7*a - 1, 5*a - 5, -7*a - 1]
 
         TESTS:
 
@@ -5871,11 +5872,16 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
             sage: K.<a> = NumberField(x^2 + 1)
             sage: K.factor(1/3)
             (Fractional ideal (3))^-1
-            sage: K.factor(1+a)
-            Fractional ideal (a + 1)
-            sage: K.factor(1+a/5)
-            (Fractional ideal (a + 1)) * (Fractional ideal (-a - 2))^-1
-              * (Fractional ideal (2*a + 1))^-1 * (Fractional ideal (-2*a + 3))
+            sage: fac = K.factor(1+a); fac # random
+            Fractional ideal (-a - 1)
+            sage: len(fac)
+            1
+            sage: fac = K.factor(1+a/5); fac # random
+            (Fractional ideal (-a - 1)) * (Fractional ideal (2*a - 1))^-1 * (Fractional ideal (-2*a - 1))^-1 * (Fractional ideal (3*a + 2))
+            sage: len(fac)
+            4
+            sage: product(I[0]^I[1] for I in list(fac))
+            Fractional ideal (1/5*a + 1)
 
         An example over a relative number field::
 
@@ -5908,9 +5914,9 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
             sage: (fi, fj) = f[::]
             sage: (fi[1], fj[1])
             (1, 1)
-            sage: fi[0] == L.fractional_ideal(1/2*a*b - a + 1/2)
+            sage: fi[0] == L.fractional_ideal(-1/2*a*b - a + 1/2)
             True
-            sage: fj[0] == L.fractional_ideal(-1/2*a*b - a + 1/2)
+            sage: fj[0] == L.fractional_ideal(1/2*a*b - a + 1/2)
             True
         """
         return self.ideal(n).factor()
@@ -6507,12 +6513,12 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
             sage: k.<a> = NumberField(x^6 + 2218926655879913714112*x^4 - 32507675650290949030789018433536*x^3 + 4923635504174417014460581055002374467948544*x^2 - 36066074010564497464129951249279114076897746988630016*x + 264187244046129768986806800244258952598300346857154900812365824)
             sage: new_basis = k.reduced_basis(prec=120)
             sage: [c.minpoly() for c in new_basis]
-            [x - 1,
-             x^2 + x + 1,
-             x^6 + 3*x^5 - 102*x^4 - 103*x^3 + 10572*x^2 - 59919*x + 127657,
-             x^6 + 3*x^5 - 102*x^4 - 103*x^3 + 10572*x^2 - 59919*x + 127657,
-             x^3 - 171*x + 848,
-             x^6 + 171*x^4 + 1696*x^3 + 29241*x^2 + 145008*x + 719104]
+            [x^2 + x + 1,
+             x - 1,
+             x^6 + 3*x^5 - 102*x^4 - 315*x^3 + 10254*x^2 + 80955*x + 198147,
+             x^6 + 213*x^4 + 12567*x^2 + 198147,
+             x^6 + 171*x^4 - 1696*x^3 + 29241*x^2 - 145008*x + 719104,
+             x^6 + 171*x^4 - 1696*x^3 + 29241*x^2 - 145008*x + 719104]
             sage: R = k.order(new_basis)
             sage: R.discriminant()==k.discriminant()
             True
@@ -7107,14 +7113,14 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
             sage: K.units(proof=True)  # takes forever, not tested
             ...
             sage: K.units(proof=False)  # result not independently verified
-            (-a^9 - a + 1,
+            (a^9 + a - 1,
+             -a^15 + a^12 - a^10 + a^9 + 2*a^8 - 3*a^7 - a^6 + 3*a^5 - a^4 - 4*a^3 + 3*a^2 + 2*a - 2,
+             a^15 + a^14 + a^13 + a^12 + a^10 - a^7 - a^6 - a^2 - 1,
+             2*a^16 - 3*a^15 + 3*a^14 - 3*a^13 + 3*a^12 - a^11 + a^9 - 3*a^8 + 4*a^7 - 5*a^6 + 6*a^5 - 4*a^4 + 3*a^3 - 2*a^2 - 2*a + 4,
              -a^16 + a^15 - a^14 + a^12 - a^11 + a^10 + a^8 - a^7 + 2*a^6 - a^4 + 3*a^3 - 2*a^2 + 2*a - 1,
-             2*a^16 - a^14 - a^13 + 3*a^12 - 2*a^10 + a^9 + 3*a^8 - 3*a^6 + 3*a^5 + 3*a^4 - 2*a^3 - 2*a^2 + 3*a + 4,
-             a^15 + a^14 + 2*a^11 + a^10 - a^9 + a^8 + 2*a^7 - a^5 + 2*a^3 - a^2 - 3*a + 1,
-             -a^16 - a^15 - a^14 - a^13 - a^12 - a^11 - a^10 - a^9 - a^8 - a^7 - a^6 - a^5 - a^4 - a^3 - a^2 + 2,
-             -2*a^16 + 3*a^15 - 3*a^14 + 3*a^13 - 3*a^12 + a^11 - a^9 + 3*a^8 - 4*a^7 + 5*a^6 - 6*a^5 + 4*a^4 - 3*a^3 + 2*a^2 + 2*a - 4,
-             a^15 - a^12 + a^10 - a^9 - 2*a^8 + 3*a^7 + a^6 - 3*a^5 + a^4 + 4*a^3 - 3*a^2 - 2*a + 2,
-             2*a^16 + a^15 - a^11 - 3*a^10 - 4*a^9 - 4*a^8 - 4*a^7 - 5*a^6 - 7*a^5 - 8*a^4 - 6*a^3 - 5*a^2 - 6*a - 7)
+             a^16 - 2*a^15 - 2*a^13 - a^12 - a^11 - 2*a^10 + a^9 - 2*a^8 + 2*a^7 - 3*a^6 - 3*a^4 - 2*a^3 - a^2 - 4*a + 2,
+             -a^15 - a^14 - 2*a^11 - a^10 + a^9 - a^8 - 2*a^7 + a^5 - 2*a^3 + a^2 + 3*a - 1,
+             -3*a^16 - 3*a^15 - 3*a^14 - 3*a^13 - 3*a^12 - 2*a^11 - 2*a^10 - 2*a^9 - a^8 + a^7 + 2*a^6 + 3*a^5 + 3*a^4 + 4*a^3 + 6*a^2 + 8*a + 8)
 
         TESTS:
 
@@ -7123,7 +7129,7 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
 
             sage: K.<a> = NumberField(1/2*x^2 - 1/6)
             sage: K.units()
-            (3*a - 2,)
+            (-3*a + 2,)
         """
         proof = proof_flag(proof)
 
@@ -7205,14 +7211,14 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
             (u0, u1, u2, u3, u4, u5, u6, u7, u8)
             sage: U.gens_values()  # result not independently verified
             [-1,
-             -a^9 - a + 1,
+             a^9 + a - 1,
+             -a^15 + a^12 - a^10 + a^9 + 2*a^8 - 3*a^7 - a^6 + 3*a^5 - a^4 - 4*a^3 + 3*a^2 + 2*a - 2,
+             a^15 + a^14 + a^13 + a^12 + a^10 - a^7 - a^6 - a^2 - 1,
+             2*a^16 - 3*a^15 + 3*a^14 - 3*a^13 + 3*a^12 - a^11 + a^9 - 3*a^8 + 4*a^7 - 5*a^6 + 6*a^5 - 4*a^4 + 3*a^3 - 2*a^2 - 2*a + 4,
              -a^16 + a^15 - a^14 + a^12 - a^11 + a^10 + a^8 - a^7 + 2*a^6 - a^4 + 3*a^3 - 2*a^2 + 2*a - 1,
-             2*a^16 - a^14 - a^13 + 3*a^12 - 2*a^10 + a^9 + 3*a^8 - 3*a^6 + 3*a^5 + 3*a^4 - 2*a^3 - 2*a^2 + 3*a + 4,
-             a^15 + a^14 + 2*a^11 + a^10 - a^9 + a^8 + 2*a^7 - a^5 + 2*a^3 - a^2 - 3*a + 1,
-             -a^16 - a^15 - a^14 - a^13 - a^12 - a^11 - a^10 - a^9 - a^8 - a^7 - a^6 - a^5 - a^4 - a^3 - a^2 + 2,
-             -2*a^16 + 3*a^15 - 3*a^14 + 3*a^13 - 3*a^12 + a^11 - a^9 + 3*a^8 - 4*a^7 + 5*a^6 - 6*a^5 + 4*a^4 - 3*a^3 + 2*a^2 + 2*a - 4,
-             a^15 - a^12 + a^10 - a^9 - 2*a^8 + 3*a^7 + a^6 - 3*a^5 + a^4 + 4*a^3 - 3*a^2 - 2*a + 2,
-             2*a^16 + a^15 - a^11 - 3*a^10 - 4*a^9 - 4*a^8 - 4*a^7 - 5*a^6 - 7*a^5 - 8*a^4 - 6*a^3 - 5*a^2 - 6*a - 7]
+             a^16 - 2*a^15 - 2*a^13 - a^12 - a^11 - 2*a^10 + a^9 - 2*a^8 + 2*a^7 - 3*a^6 - 3*a^4 - 2*a^3 - a^2 - 4*a + 2,
+             -a^15 - a^14 - 2*a^11 - a^10 + a^9 - a^8 - 2*a^7 + a^5 - 2*a^3 + a^2 + 3*a - 1,
+             -3*a^16 - 3*a^15 - 3*a^14 - 3*a^13 - 3*a^12 - 2*a^11 - 2*a^10 - 2*a^9 - a^8 + a^7 + 2*a^6 + 3*a^5 + 3*a^4 + 4*a^3 + 6*a^2 + 8*a + 8]
         """
         proof = proof_flag(proof)
 
@@ -7261,8 +7267,8 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
             sage: U = K.S_unit_group(S=a); U
             S-unit group with structure C10 x Z x Z x Z of
              Number Field in a with defining polynomial x^4 - 10*x^3 + 100*x^2 - 375*x + 1375
-             with S = (Fractional ideal (5, 1/275*a^3 + 4/55*a^2 - 5/11*a + 5),
-                       Fractional ideal (11, 1/275*a^3 + 4/55*a^2 - 5/11*a + 9))
+             with S = (Fractional ideal (5, -7/275*a^3 + 1/11*a^2 - 9/11*a),
+                       Fractional ideal (11, -7/275*a^3 + 1/11*a^2 - 9/11*a + 3))
             sage: U.gens()
             (u0, u1, u2, u3)
             sage: U.gens_values()  # random
@@ -7273,8 +7279,8 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
             sage: [u.multiplicative_order() for u in U.gens()]
             [10, +Infinity, +Infinity, +Infinity]
             sage: U.primes()
-            (Fractional ideal (5, 1/275*a^3 + 4/55*a^2 - 5/11*a + 5),
-             Fractional ideal (11, 1/275*a^3 + 4/55*a^2 - 5/11*a + 9))
+            (Fractional ideal (5, -7/275*a^3 + 1/11*a^2 - 9/11*a),
+             Fractional ideal (11, -7/275*a^3 + 1/11*a^2 - 9/11*a + 3))
 
         With the default value of `S`, the S-unit group is the same as
         the global unit group::
@@ -7426,8 +7432,8 @@ class NumberField_generic(WithEqualityById, number_field_base.NumberField):
 
             sage: # needs sage.rings.padics
             sage: solutions, bound = K.S_unit_solutions(S, prec=100, include_bound=True)
-            sage: bound
-            7
+            sage: bound in (6, 7)
+            True
         """
         from .S_unit_solver import solve_S_unit_equation
         return solve_S_unit_equation(self, S, prec, include_exponents, include_bound, proof)
@@ -8782,7 +8788,7 @@ class NumberField_absolute(NumberField_generic):
             (Number Field in a1 with defining polynomial x^2 - 2, Ring morphism:
               From: Number Field in a1 with defining polynomial x^2 - 2
               To:   Number Field in a with defining polynomial 2*x^4 + 6*x^2 + 1/2
-              Defn: a1 |--> a^2 + 3/2, None),
+              Defn: a1 |--> -a^2 - 3/2, None),
             (Number Field in a2 with defining polynomial x^2 + 4, Ring morphism:
               From: Number Field in a2 with defining polynomial x^2 + 4
               To:   Number Field in a with defining polynomial 2*x^4 + 6*x^2 + 1/2
@@ -8790,14 +8796,14 @@ class NumberField_absolute(NumberField_generic):
             (Number Field in a3 with defining polynomial x^2 + 2, Ring morphism:
               From: Number Field in a3 with defining polynomial x^2 + 2
               To:   Number Field in a with defining polynomial 2*x^4 + 6*x^2 + 1/2
-              Defn: a3 |--> 2*a^3 + 5*a, None),
+              Defn: a3 |--> -2*a^3 - 5*a, None),
             (Number Field in a4 with defining polynomial x^4 + 1, Ring morphism:
               From: Number Field in a4 with defining polynomial x^4 + 1
               To:   Number Field in a with defining polynomial 2*x^4 + 6*x^2 + 1/2
-              Defn: a4 |--> a^3 + 1/2*a^2 + 5/2*a + 3/4, Ring morphism:
+              Defn: a4 |--> -a^3 - 1/2*a^2 - 5/2*a - 3/4, Ring morphism:
               From: Number Field in a with defining polynomial 2*x^4 + 6*x^2 + 1/2
               To:   Number Field in a4 with defining polynomial x^4 + 1
-              Defn: a |--> -1/2*a4^3 + a4^2 - 1/2*a4)
+              Defn: a |--> 1/2*a4^3 + a4^2 + 1/2*a4)
             ]
         """
         return self._subfields_helper(degree=degree, name=name,
@@ -12694,12 +12700,12 @@ def _splitting_classes_gens_(K, m, d):
         sage: L = K.subfields(20)[0][0]
         sage: L.conductor()                                                             # needs sage.groups
         101
-        sage: _splitting_classes_gens_(L,101,20)                                        # needs sage.libs.gap  # optional - gap_package_polycyclic
+        sage: _splitting_classes_gens_(L,101,20)        # optional - gap_package_polycyclic, needs sage.libs.gap
         [95]
 
         sage: K = CyclotomicField(44)
         sage: L = K.subfields(4)[0][0]
-        sage: _splitting_classes_gens_(L,44,4)                                          # needs sage.libs.gap  # optional - gap_package_polycyclic
+        sage: _splitting_classes_gens_(L,44,4)  # optional - gap_package_polycyclic, needs sage.libs.gap
         [37]
 
         sage: K = CyclotomicField(44)
@@ -12711,7 +12717,7 @@ def _splitting_classes_gens_(K, m, d):
          with zeta44_0 = 3.837971894457990?
         sage: L.conductor()                                                             # needs sage.groups
         11
-        sage: _splitting_classes_gens_(L,11,5)                                          # needs sage.libs.gap  # optional - gap_package_polycyclic
+        sage: _splitting_classes_gens_(L,11,5)  # optional - gap_package_polycyclic, needs sage.libs.gap
         [10]
     """
     from sage.groups.abelian_gps.abelian_group import AbelianGroup
diff --git a/src/sage/rings/number_field/number_field_element.pyx b/src/sage/rings/number_field/number_field_element.pyx
index 27432813b2b..1a676ee087b 100644
--- a/src/sage/rings/number_field/number_field_element.pyx
+++ b/src/sage/rings/number_field/number_field_element.pyx
@@ -1954,14 +1954,14 @@ cdef class NumberFieldElement(NumberFieldElement_base):
             sage: x = polygen(ZZ, 'x')
             sage: K.<i> = NumberField(x^2 + 1)
             sage: (6*i + 6).factor()
-            (-i) * (i + 1)^3 * 3
+            (i) * (-i - 1)^3 * 3
 
         In the following example, the class number is 2.  If a factorization
         in prime elements exists, we will find it::
 
             sage: K.<a> = NumberField(x^2 - 10)
             sage: factor(169*a + 531)
-            (-6*a - 19) * (-3*a - 1) * (-2*a + 9)
+            (-6*a - 19) * (2*a - 9) * (3*a + 1)
             sage: factor(K(3))
             Traceback (most recent call last):
             ...
@@ -4238,7 +4238,7 @@ cdef class NumberFieldElement(NumberFieldElement_base):
 
             sage: P5s = F(5).support()
             sage: P5s
-            [Fractional ideal (-t^2 - 1), Fractional ideal (t^2 - 2*t - 1)]
+            [Fractional ideal (t^2 + 1), Fractional ideal (t^2 - 2*t - 1)]
             sage: all(5 in P5 for P5 in P5s)
             True
             sage: all(P5.is_prime() for P5 in P5s)
@@ -4487,7 +4487,7 @@ cdef class NumberFieldElement(NumberFieldElement_base):
             sage: f = Qi.embeddings(K)[0]
             sage: a = f(2+3*i) * (2-zeta)^2
             sage: a.descend_mod_power(Qi,2)
-            [-2*i + 3, 3*i + 2]
+            [3*i + 2, 2*i - 3]
 
         An absolute example::
 
diff --git a/src/sage/rings/number_field/number_field_ideal.py b/src/sage/rings/number_field/number_field_ideal.py
index d62ae34fb95..72e14adb3af 100644
--- a/src/sage/rings/number_field/number_field_ideal.py
+++ b/src/sage/rings/number_field/number_field_ideal.py
@@ -75,7 +75,7 @@ class NumberFieldIdeal(Ideal_generic):
         Fractional ideal (3)
         sage: F = pari(K).idealprimedec(5)
         sage: K.ideal(F[0])
-        Fractional ideal (2*i + 1)
+        Fractional ideal (-2*i - 1)
 
     TESTS:
 
@@ -236,7 +236,7 @@ class NumberFieldIdeal(Ideal_generic):
             sage: K.<a> = NumberField(x^2 + 3); K
             Number Field in a with defining polynomial x^2 + 3
             sage: f = K.factor(15); f
-            (Fractional ideal (1/2*a + 3/2))^2 * (Fractional ideal (5))
+            (Fractional ideal (-a))^2 * (Fractional ideal (5))
             sage: (f[0][0] < f[1][0])
             True
             sage: (f[0][0] == f[0][0])
@@ -273,7 +273,7 @@ class NumberFieldIdeal(Ideal_generic):
             sage: A = K.ideal([5, 2 + I])
             sage: B = K.ideal([13, 5 + 12*I])
             sage: A*B
-            Fractional ideal (4*I - 7)
+            Fractional ideal (-4*I + 7)
             sage: (K.ideal(3 + I) * K.ideal(7 + I)).gens()
             (10*I + 20,)
 
@@ -683,17 +683,17 @@ class NumberFieldIdeal(Ideal_generic):
             sage: I.free_module()
             Free module of degree 4 and rank 4 over Integer Ring
             User basis matrix:
-            [  4   0   0   0]
-            [ -3   7  -1   1]
-            [  3   7   1   1]
-            [  0 -10   0  -2]
+            [ 4  0  0  0]
+            [ 3  7  1  1]
+            [ 0 10  0  2]
+            [ 3 -7  1 -1]
             sage: J = I^(-1); J.free_module()
             Free module of degree 4 and rank 4 over Integer Ring
             User basis matrix:
             [  1/4     0     0     0]
-            [-3/16  7/16 -1/16  1/16]
             [ 3/16  7/16  1/16  1/16]
-            [    0  -5/8     0  -1/8]
+            [    0   5/8     0   1/8]
+            [ 3/16 -7/16  1/16 -1/16]
 
         An example of intersecting ideals by intersecting free modules.::
 
@@ -790,7 +790,7 @@ class NumberFieldIdeal(Ideal_generic):
             sage: J.is_principal()
             False
             sage: J.gens_reduced()
-            (5, a)
+            (5, -a)
             sage: all(j.parent() is K for j in J.gens())
             True
             sage: all(j.parent() is K for j in J.gens_reduced())
@@ -2069,7 +2069,7 @@ class NumberFieldFractionalIdeal(MultiplicativeGroupElement, NumberFieldIdeal, I
             sage: K.<a> = NumberField(x^2 + 2); K
             Number Field in a with defining polynomial x^2 + 2
             sage: f = K.factor(2); f
-            (Fractional ideal (a))^2
+            (Fractional ideal (...a))^2
             sage: f[0][0].ramification_index()
             2
             sage: K.ideal(13).ramification_index()
@@ -2416,9 +2416,9 @@ class NumberFieldFractionalIdeal(MultiplicativeGroupElement, NumberFieldIdeal, I
             sage: I = K.ideal((3+4*i)/5); I
             Fractional ideal (4/5*i + 3/5)
             sage: I.denominator()
-            Fractional ideal (2*i + 1)
+            Fractional ideal (-2*i - 1)
             sage: I.numerator()
-            Fractional ideal (-i - 2)
+            Fractional ideal (2*i - 1)
             sage: I.numerator().is_integral() and I.denominator().is_integral()
             True
             sage: I.numerator() + I.denominator() == K.unit_ideal()
@@ -2447,9 +2447,9 @@ class NumberFieldFractionalIdeal(MultiplicativeGroupElement, NumberFieldIdeal, I
             sage: I = K.ideal((3+4*i)/5); I
             Fractional ideal (4/5*i + 3/5)
             sage: I.denominator()
-            Fractional ideal (2*i + 1)
+            Fractional ideal (-2*i - 1)
             sage: I.numerator()
-            Fractional ideal (-i - 2)
+            Fractional ideal (2*i - 1)
             sage: I.numerator().is_integral() and I.denominator().is_integral()
             True
             sage: I.numerator() + I.denominator() == K.unit_ideal()
@@ -3164,11 +3164,11 @@ class NumberFieldFractionalIdeal(MultiplicativeGroupElement, NumberFieldIdeal, I
             Partially defined quotient map
              from Number Field in i with defining polynomial x^2 + 1
                to an explicit vector space representation for the quotient of
-                  the ring of integers by (p,I) for the ideal I=Fractional ideal (-i - 2).
+                  the ring of integers by (p,I) for the ideal I=Fractional ideal (2*i - 1).
             sage: lift
             Lifting map
                to Gaussian Integers generated by i in Number Field in i with defining polynomial x^2 + 1
-               from quotient of integers by Fractional ideal (-i - 2)
+               from quotient of integers by Fractional ideal (2*i - 1)
         """
         return quotient_char_p(self, p)
 
@@ -3213,11 +3213,11 @@ class NumberFieldFractionalIdeal(MultiplicativeGroupElement, NumberFieldIdeal, I
 
             sage: K.<i> = NumberField(x^2 + 1)
             sage: P1, P2 = [g[0] for g in K.factor(5)]; P1, P2
-            (Fractional ideal (-i - 2), Fractional ideal (2*i + 1))
+            (Fractional ideal (2*i - 1), Fractional ideal (-2*i - 1))
             sage: a = 1/(1+2*i)
             sage: F1, F2 = [g.residue_field() for g in [P1, P2]]; F1, F2
-            (Residue field of Fractional ideal (-i - 2),
-             Residue field of Fractional ideal (2*i + 1))
+            (Residue field of Fractional ideal (2*i - 1),
+             Residue field of Fractional ideal (-2*i - 1))
             sage: a.valuation(P1)
             0
             sage: F1(i/7)
@@ -3230,7 +3230,7 @@ class NumberFieldFractionalIdeal(MultiplicativeGroupElement, NumberFieldIdeal, I
             Traceback (most recent call last):
             ...
             ZeroDivisionError: Cannot reduce field element -2/5*i + 1/5
-            modulo Fractional ideal (2*i + 1): it has negative valuation
+            modulo Fractional ideal (-2*i - 1): it has negative valuation
 
         An example with a relative number field::
 
@@ -3491,7 +3491,7 @@ def quotient_char_p(I, p):
         []
 
         sage: I = K.factor(13)[0][0]; I
-        Fractional ideal (-2*i + 3)
+        Fractional ideal (3*i + 2)
         sage: I.residue_class_degree()
         1
         sage: quotient_char_p(I, 13)[0]
diff --git a/src/sage/rings/number_field/number_field_ideal_rel.py b/src/sage/rings/number_field/number_field_ideal_rel.py
index 7f6cfd9b1b7..0bae4e74e13 100644
--- a/src/sage/rings/number_field/number_field_ideal_rel.py
+++ b/src/sage/rings/number_field/number_field_ideal_rel.py
@@ -11,7 +11,7 @@ EXAMPLES::
     sage: G = [from_A(z) for z in I.gens()]; G
     [7, -2*b*a - 1]
     sage: K.fractional_ideal(G)
-    Fractional ideal ((1/2*b + 2)*a - 1/2*b + 2)
+    Fractional ideal ((-1/2*b + 2)*a - 1/2*b - 2)
     sage: K.fractional_ideal(G).absolute_norm().factor()
     7^2
 
@@ -277,7 +277,7 @@ class NumberFieldFractionalIdeal_rel(NumberFieldFractionalIdeal):
             sage: L.<b> = K.extension(5*x^2 + 1)
             sage: P = L.primes_above(2)[0]
             sage: P.gens_reduced()
-            (2, -15*a*b + 3*a + 1)
+            (2, -15*a*b - 3*a + 1)
         """
         try:
             # Compute the single generator, if it exists
@@ -385,7 +385,7 @@ class NumberFieldFractionalIdeal_rel(NumberFieldFractionalIdeal):
             sage: K.<a> = NumberField(x^2 + 6)
             sage: L.<b> = K.extension(K['x'].gen()^4 + a)
             sage: N = L.ideal(b).relative_norm(); N
-            Fractional ideal (-a)
+            Fractional ideal (a)
             sage: N.parent()
             Monoid of ideals of Number Field in a with defining polynomial x^2 + 6
             sage: N.ring()
@@ -410,7 +410,7 @@ class NumberFieldFractionalIdeal_rel(NumberFieldFractionalIdeal):
             sage: L.<b> = K.extension(5*x^2 + 1)
             sage: P = L.primes_above(2)[0]
             sage: P.relative_norm()
-            Fractional ideal (6*a + 2)
+            Fractional ideal (6*a - 2)
         """
         L = self.number_field()
         K = L.base_field()
@@ -529,7 +529,7 @@ class NumberFieldFractionalIdeal_rel(NumberFieldFractionalIdeal):
             sage: L.<b> = K.extension(5*x^2 + 1)
             sage: P = L.primes_above(2)[0]
             sage: P.ideal_below()
-            Fractional ideal (6*a + 2)
+            Fractional ideal (6*a - 2)
         """
         L = self.number_field()
         K = L.base_field()
@@ -548,14 +548,12 @@ class NumberFieldFractionalIdeal_rel(NumberFieldFractionalIdeal):
             sage: x = polygen(ZZ, 'x')
             sage: K.<a, b> = QQ.extension([x^2 + 11, x^2 - 5])
             sage: K.factor(5)
-            (Fractional ideal (5, (-1/4*b - 1/4)*a + 1/4*b - 3/4))^2
-             * (Fractional ideal (5, (-1/4*b - 1/4)*a + 1/4*b - 7/4))^2
+            (Fractional ideal (5, (1/4*b - 1/4)*a + 1/4*b + 3/4))^2 * (Fractional ideal (5, (1/4*b - 1/4)*a + 1/4*b + 7/4))^2
             sage: K.ideal(5).factor()
-            (Fractional ideal (5, (-1/4*b - 1/4)*a + 1/4*b - 3/4))^2
-             * (Fractional ideal (5, (-1/4*b - 1/4)*a + 1/4*b - 7/4))^2
+            (Fractional ideal (5, (1/4*b - 1/4)*a + 1/4*b + 3/4))^2 * (Fractional ideal (5, (1/4*b - 1/4)*a + 1/4*b + 7/4))^2
             sage: K.ideal(5).prime_factors()
-            [Fractional ideal (5, (-1/4*b - 1/4)*a + 1/4*b - 3/4),
-             Fractional ideal (5, (-1/4*b - 1/4)*a + 1/4*b - 7/4)]
+            [Fractional ideal (5, (1/4*b - 1/4)*a + 1/4*b + 3/4),
+             Fractional ideal (5, (1/4*b - 1/4)*a + 1/4*b + 7/4)]
 
             sage: PQ.<X> = QQ[]
             sage: F.<a, b> = NumberFieldTower([X^2 - 2, X^2 - 3])
@@ -914,7 +912,7 @@ def is_NumberFieldFractionalIdeal_rel(x):
         sage: is_NumberFieldFractionalIdeal_rel(I)
         True
         sage: N = I.relative_norm(); N
-        Fractional ideal (-a)
+        Fractional ideal (a)
         sage: is_NumberFieldFractionalIdeal_rel(N)
         False
     """
diff --git a/src/sage/rings/number_field/number_field_rel.py b/src/sage/rings/number_field/number_field_rel.py
index 9ab6fd4c261..6e9bf71baf4 100644
--- a/src/sage/rings/number_field/number_field_rel.py
+++ b/src/sage/rings/number_field/number_field_rel.py
@@ -233,21 +233,21 @@ class NumberField_relative(NumberField_generic):
             sage: l.<b> = k.extension(5*x^2 + 3); l
             Number Field in b with defining polynomial 5*x^2 + 3 over its base field
             sage: l.pari_rnf()
-            [x^2 + (-y^3 + 1/2*y^2 - 6*y + 3/2)*x + (-3/4*y^3 - 1/4*y^2 - 17/4*y - 19/4), ..., y^4 + 6*y^2 + 1, x^2 + (-y^3 + 1/2*y^2 - 6*y + 3/2)*x + (-3/4*y^3 - 1/4*y^2 - 17/4*y - 19/4)], [0, 0]]
+            [x^2 + (5/4*y^3 - 1/4*y^2 + 27/4*y - 3/4)*x + (-9/4*y^3 - 1/4*y^2 - 47/4*y - 7/4), ..., y^4 + 6*y^2 + 1, x^2 + (5/4*y^3 - 1/4*y^2 + 27/4*y - 3/4)*x + (-9/4*y^3 - 1/4*y^2 - 47/4*y - 7/4)], [0, 0]]
             sage: b
             b
 
             sage: l.<b> = k.extension(x^2 + 3/5); l
             Number Field in b with defining polynomial x^2 + 3/5 over its base field
             sage: l.pari_rnf()
-            [x^2 + (-y^3 + 1/2*y^2 - 6*y + 3/2)*x + (-3/4*y^3 - 1/4*y^2 - 17/4*y - 19/4), ..., y^4 + 6*y^2 + 1, x^2 + (-y^3 + 1/2*y^2 - 6*y + 3/2)*x + (-3/4*y^3 - 1/4*y^2 - 17/4*y - 19/4)], [0, 0]]
+            [x^2 + (5/4*y^3 - 1/4*y^2 + 27/4*y - 3/4)*x + (-9/4*y^3 - 1/4*y^2 - 47/4*y - 7/4), ..., y^4 + 6*y^2 + 1, x^2 + (5/4*y^3 - 1/4*y^2 + 27/4*y - 3/4)*x + (-9/4*y^3 - 1/4*y^2 - 47/4*y - 7/4)], [0, 0]]
             sage: b
             b
 
             sage: l.<b> = k.extension(x - 1/a0); l
             Number Field in b with defining polynomial x + 1/2*a0 over its base field
             sage: l.pari_rnf()
-            [x, [4, -x^3 - x^2 - 7*x - 3, -x^3 + x^2 - 7*x + 3, 2*x^3 + 10*x], ..., [x^4 + 6*x^2 + 1, -x, -1, y^4 + 6*y^2 + 1, x], [0, 0]]
+            [x, [4, -x^3 + x^2 - 7*x + 3, -2*x^3 - 10*x, x^3 + x^2 + 7*x + 3], ..., [x^4 + 6*x^2 + 1, -x, -1, y^4 + 6*y^2 + 1, x], [0, 0]]
             sage: b
             -1/2*a0
 
@@ -1635,9 +1635,9 @@ class NumberField_relative(NumberField_generic):
             sage: K.<a> = NumberField(x^2 + 1)
             sage: L.<b> = K.extension(x^2 - 1/2)
             sage: L._pari_relative_structure()
-            (x^2 + Mod(-y, y^2 + 1),
-             Mod(Mod(1/2*y - 1/2, y^2 + 1)*x, x^2 + Mod(-y, y^2 + 1)),
-             Mod(Mod(-y - 1, y^2 + 1)*x, Mod(1, y^2 + 1)*x^2 + Mod(-1/2, y^2 + 1)))
+            (x^2 + Mod(y, y^2 + 1),
+             Mod(Mod(-1/2*y - 1/2, y^2 + 1)*x, x^2 + Mod(y, y^2 + 1)),
+             Mod(Mod(y - 1, y^2 + 1)*x, x^2 + Mod(-1/2, y^2 + 1)))
 
         An example where both fields are defined by non-integral or
         non-monic polynomials::
@@ -1937,7 +1937,7 @@ class NumberField_relative(NumberField_generic):
             sage: k.relative_polynomial()
             x^2 + 1/3
             sage: k.pari_relative_polynomial()
-            x^2 + Mod(y, y^2 + 1)*x - 1
+            x^2 + Mod(-y, y^2 + 1)*x - 1
         """
         return QQ['x'](self._pari_rnfeq()[0])
 
@@ -2721,7 +2721,7 @@ class NumberField_relative(NumberField_generic):
             sage: x = polygen(ZZ, 'x')
             sage: K.<a, b> = NumberField([x^2 + 23, x^2 - 3])
             sage: P = K.prime_factors(5)[0]; P
-            Fractional ideal (5, 1/2*a + b - 5/2)
+            Fractional ideal (5, -1/2*a + b + 5/2)
             sage: u = K.uniformizer(P)
             sage: u.valuation(P)
             1
diff --git a/src/sage/rings/number_field/order.py b/src/sage/rings/number_field/order.py
index e54d2f454af..85fb330d254 100644
--- a/src/sage/rings/number_field/order.py
+++ b/src/sage/rings/number_field/order.py
@@ -495,7 +495,7 @@ class Order(IntegralDomain, sage.rings.abc.Order):
             sage: K.<a> = NumberField(x^2 + 2)
             sage: R = K.maximal_order()
             sage: R.fractional_ideal(2/3 + 7*a, a)
-            Fractional ideal (1/3*a)
+            Fractional ideal (-1/3*a)
         """
         return self.number_field().fractional_ideal(*args, **kwds)
 
@@ -569,7 +569,7 @@ class Order(IntegralDomain, sage.rings.abc.Order):
             sage: k.<a> = NumberField(x^2 + 5077); G = k.class_group(); G
             Class group of order 22 with structure C22 of Number Field in a with defining polynomial x^2 + 5077
             sage: G.0 ^ -9
-            Fractional ideal class (43, a + 13)
+            Fractional ideal class (67, a + 45)
             sage: Ok = k.maximal_order(); Ok
             Maximal Order generated by a in Number Field in a with defining polynomial x^2 + 5077
             sage: Ok * (11, a + 7)
@@ -2918,7 +2918,7 @@ def GaussianIntegers(names='I', latex_name='i'):
         sage: ZZI
         Gaussian Integers generated by I in Number Field in I with defining polynomial x^2 + 1 with I = 1*I
         sage: factor(3 + I)
-        (-I) * (I + 1) * (2*I + 1)
+        (-I) * (I - 2) * (-I + 1)
         sage: CC(I)
         1.00000000000000*I
         sage: I.minpoly()
@@ -2949,7 +2949,7 @@ def EisensteinIntegers(names='omega'):
          with defining polynomial x^2 + x + 1
          with omega = -0.50000000000000000? + 0.866025403784439?*I
         sage: factor(3 + omega)
-        (-1) * (-omega - 3)
+        (omega + 1) * (-2*omega + 1)
         sage: CC(omega)
         -0.500000000000000 + 0.866025403784439*I
         sage: omega.minpoly()
diff --git a/src/sage/rings/number_field/selmer_group.py b/src/sage/rings/number_field/selmer_group.py
index acdf2954fdf..7da6b58a209 100644
--- a/src/sage/rings/number_field/selmer_group.py
+++ b/src/sage/rings/number_field/selmer_group.py
@@ -71,7 +71,7 @@ def _ideal_generator(I):
 
         sage: K.<a> = QuadraticField(-11)
         sage: [_ideal_generator(K.prime_above(p)) for p in primes(25)]
-        [2, 1/2*a - 1/2, -1/2*a - 3/2, 7, -a, 13, 17, 19, 1/2*a + 9/2]
+        [2, 1/2*a - 1/2, -1/2*a - 3/2, 7, a, 13, 17, 19, 1/2*a + 9/2]
     """
     try:
         return I.gens_reduced()[0]
diff --git a/src/sage/rings/polynomial/polynomial_quotient_ring.py b/src/sage/rings/polynomial/polynomial_quotient_ring.py
index e12977e8464..c5bf39a1c73 100644
--- a/src/sage/rings/polynomial/polynomial_quotient_ring.py
+++ b/src/sage/rings/polynomial/polynomial_quotient_ring.py
@@ -1429,13 +1429,13 @@ class PolynomialQuotientRing_generic(QuotientRing_generic):
             sage: R.<x> = K[]
             sage: S.<xbar> = R.quotient(x^2 + 23)
             sage: S.S_class_group([])
-            [((2, -a + 1, 1/2*xbar + 1/2, -1/2*a*xbar + 1/2*a + 1), 6)]
+            [((2, a + 1, -1/2*xbar + 3/2, 1/2*a*xbar - 1/2*a + 1), 6)]
             sage: S.S_class_group([K.ideal(3, a-1)])
             []
             sage: S.S_class_group([K.ideal(2, a+1)])
             []
             sage: S.S_class_group([K.ideal(a)])
-            [((2, -a + 1, 1/2*xbar + 1/2, -1/2*a*xbar + 1/2*a + 1), 6)]
+            [((2, a + 1, -1/2*xbar + 3/2, 1/2*a*xbar - 1/2*a + 1), 6)]
 
         Now we take an example over a nontrivial base with two factors, each
         contributing to the class group::
@@ -1495,14 +1495,14 @@ class PolynomialQuotientRing_generic(QuotientRing_generic):
             sage: C = S.S_class_group([])
             sage: C[:2]
             [((1/4*xbar^2 + 31/4,
-               (-1/8*a + 1/8)*xbar^2 - 31/8*a + 31/8,
-               1/16*xbar^3 + 1/16*xbar^2 + 31/16*xbar + 31/16,
-               -1/16*a*xbar^3 + (1/16*a + 1/8)*xbar^2 - 31/16*a*xbar + 31/16*a + 31/8),
+               (1/8*a + 1/8)*xbar^2 + 31/8*a + 31/8,
+               -1/16*xbar^3 + 3/16*xbar^2 - 31/16*xbar + 93/16,
+               1/16*a*xbar^3 + (-1/16*a + 1/8)*xbar^2 + 31/16*a*xbar - 31/16*a + 31/8),
               6),
              ((-1/4*xbar^2 - 23/4,
-               (1/8*a - 1/8)*xbar^2 + 23/8*a - 23/8,
-               -1/16*xbar^3 - 1/16*xbar^2 - 23/16*xbar - 23/16,
-               1/16*a*xbar^3 + (-1/16*a - 1/8)*xbar^2 + 23/16*a*xbar - 23/16*a - 23/8),
+               (-1/8*a - 1/8)*xbar^2 - 23/8*a - 23/8,
+               1/16*xbar^3 + 1/16*xbar^2 + 23/16*xbar + 23/16,
+               -1/16*a*xbar^3 + (1/16*a - 1/8)*xbar^2 - 23/16*a*xbar + 23/16*a - 23/8),
               6)]
             sage: C[2][1]
             2
@@ -1514,11 +1514,11 @@ class PolynomialQuotientRing_generic(QuotientRing_generic):
             ....:     1/16*a*xbar^3 - 1/16*a*xbar^2 + 23/16*a*xbar - 23/16*a)
             sage: gens[0] == expected_gens[0]
             True
-            sage: gens[1] in (expected_gens[1], expected_gens[1]/2 + expected_gens[0]/2)
+            sage: gens[1] in (expected_gens[1], expected_gens[1]/2 + expected_gens[0]/2, -expected_gens[1]/2 + expected_gens[0]/2)
             True
-            sage: gens[2] in (expected_gens[2], expected_gens[2] + expected_gens[0]/2)
+            sage: gens[2] in (expected_gens[2], expected_gens[2] + expected_gens[0]/2, -expected_gens[2] + expected_gens[0]/2)
             True
-            sage: gens[3] in (expected_gens[3], expected_gens[3] + expected_gens[0]/2)
+            sage: gens[3] in (expected_gens[3], expected_gens[3] + expected_gens[0]/2, -expected_gens[3] + expected_gens[0]/2)
             True
         """
         fields, isos, iso_classes = self._S_decomposition(tuple(S))
@@ -1611,7 +1611,7 @@ class PolynomialQuotientRing_generic(QuotientRing_generic):
             sage: R.<x> = K[]
             sage: S.<xbar> = R.quotient(x^2 + 23)
             sage: S.class_group()
-            [((2, -a + 1, 1/2*xbar + 1/2, -1/2*a*xbar + 1/2*a + 1), 6)]
+            [((2, a + 1, -1/2*xbar + 3/2, 1/2*a*xbar - 1/2*a + 1), 6)]
 
         Here is an example of a product of number fields, both of which
         contribute to the class group::
@@ -1711,19 +1711,19 @@ class PolynomialQuotientRing_generic(QuotientRing_generic):
              with defining polynomial x^2 + 3 with a = 1.732050807568878?*I
              with modulus y^3 + 5
             sage: [u for u, o in L.S_units([]) if o is Infinity]
-            [(-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2,
-             2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]
+            [(-1/3*a - 1)*b^2 - 4/3*a*b - 4/3*a + 3,
+             (-1/3*a - 1)*b^2 + (2/3*a - 2)*b + 13/6*a - 1/2]
             sage: [u for u, o in L.S_units([K.ideal(1/2*a - 3/2)])
             ....:  if o is Infinity]
             [(-1/6*a - 1/2)*b^2 + (1/3*a - 1)*b + 4/3*a,
-             (-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2,
-             2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]
+             (-1/3*a - 1)*b^2 - 4/3*a*b - 4/3*a + 3,
+             (-1/3*a - 1)*b^2 + (2/3*a - 2)*b + 13/6*a - 1/2]
             sage: [u for u, o in L.S_units([K.ideal(2)]) if o is Infinity]
             [(1/2*a - 1/2)*b^2 + (a + 1)*b + 3,
-             (1/6*a + 1/2)*b^2 + (-1/3*a + 1)*b - 5/6*a + 1/2,
              (1/6*a + 1/2)*b^2 + (-1/3*a + 1)*b - 5/6*a - 1/2,
-             (-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2,
-             2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]
+             1/3*a*b^2 + (1/3*a + 1)*b - 1/6*a + 3/2,
+             (-1/3*a - 1)*b^2 - 4/3*a*b - 4/3*a + 3,
+             (-1/3*a - 1)*b^2 + (2/3*a - 2)*b + 13/6*a - 1/2]
 
         Note that all the returned values live where we expect them to::
 
@@ -1808,8 +1808,8 @@ class PolynomialQuotientRing_generic(QuotientRing_generic):
              with defining polynomial x^2 + 3 with a = 1.732050807568878?*I
              with modulus y^3 + 5
             sage: [u for u, o in L.units() if o is Infinity]
-            [(-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2,
-             2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]
+            [(-1/3*a - 1)*b^2 - 4/3*a*b - 4/3*a + 3,
+             (-1/3*a - 1)*b^2 + (2/3*a - 2)*b + 13/6*a - 1/2]
             sage: L.<b> = K.extension(y^3 + 5)
             sage: L.unit_group()
             Unit group with structure C6 x Z x Z of
@@ -1817,8 +1817,8 @@ class PolynomialQuotientRing_generic(QuotientRing_generic):
             sage: L.unit_group().gens()    # abstract generators
             (u0, u1, u2)
             sage: L.unit_group().gens_values()[1:]
-            [(-1/3*a - 1)*b^2 - 4/3*a*b - 5/6*a + 7/2,
-             2/3*a*b^2 + (2/3*a - 2)*b - 5/6*a - 7/2]
+            [(-1/3*a - 1)*b^2 - 4/3*a*b - 4/3*a + 3,
+             (-1/3*a - 1)*b^2 + (2/3*a - 2)*b + 13/6*a - 1/2]
 
         Note that all the returned values live where we expect them to::
 
diff --git a/src/sage/rings/qqbar.py b/src/sage/rings/qqbar.py
index 47365a9a8c6..7fd8c6f0c05 100644
--- a/src/sage/rings/qqbar.py
+++ b/src/sage/rings/qqbar.py
@@ -2776,11 +2776,12 @@ def number_field_elements_from_algebraics(numbers, minimal=False,
             To:   Algebraic Real Field
             Defn: a |--> 1.732050807568878?)
         sage: number_field_elements_from_algebraics((rt2,qqI))                          # needs sage.symbolic
-        (Number Field in a with defining polynomial y^4 + 1, [-a^3 + a, a^2],
+        (Number Field in a with defining polynomial y^4 + 1,
+         [a^3 - a, a^2],
          Ring morphism:
-            From: Number Field in a with defining polynomial y^4 + 1
-            To:   Algebraic Field
-            Defn: a |--> 0.7071067811865475? + 0.7071067811865475?*I)
+           From: Number Field in a with defining polynomial y^4 + 1
+           To:   Algebraic Field
+           Defn: a |--> -0.7071067811865475? - 0.7071067811865475?*I)
 
     Note that for the first example, where \sage does not realize that
     the number is real, we get a homomorphism to ``QQbar``::
@@ -4590,8 +4591,7 @@ class AlgebraicNumber_base(sage.structure.element.FieldElement):
             Number Field in a with defining polynomial y^4 - 20*y^2 + 81
              with a in -3.789313782671036?
             sage: (QQbar(7)^(3/5))._exact_field()
-            Number Field in a with defining polynomial
-             y^5 - 2*y^4 - 18*y^3 + 38*y^2 + 82*y - 181 with a in 2.554256611698490?
+            Number Field in a with defining polynomial y^5 - 7 with a in 1.475773161594552?
         """
         sd = self._descr
         if isinstance(sd, (ANRational, ANExtensionElement)):
@@ -4611,7 +4611,7 @@ class AlgebraicNumber_base(sage.structure.element.FieldElement):
             sage: (sqrt(QQbar(2)) + sqrt(QQbar(19)))._exact_value()
             -1/9*a^3 + a^2 + 11/9*a - 10 where a^4 - 20*a^2 + 81 = 0 and a in -3.789313782671036?
             sage: (QQbar(7)^(3/5))._exact_value()
-            2*a^4 + 2*a^3 - 34*a^2 - 17*a + 150 where a^5 - 2*a^4 - 18*a^3 + 38*a^2 + 82*a - 181 = 0 and a in 2.554256611698490?
+            a^3 where a^5 - 7 = 0 and a in 1.475773161594552?
         """
         sd = self._descr
         if isinstance(sd, (ANRational, ANExtensionElement)):
@@ -7839,8 +7839,8 @@ class ANExtensionElement(ANDescr):
             sage: sage_input(v, verify=True)
             # Verified
             R.<y> = QQ[]
-            v = QQbar.polynomial_root(AA.common_polynomial(y^8 - y^7 + y^5 - y^4 + y^3 - y + 1), CIF(RIF(RR(0.91354545764260087), RR(0.91354545764260098)), RIF(RR(0.40673664307580015), RR(0.40673664307580021))))
-            v^5 + v^3
+            v = QQbar.polynomial_root(AA.common_polynomial(y^8 - y^7 + y^5 - y^4 + y^3 - y + 1), CIF(RIF(RR(0.66913060635885813), RR(0.66913060635885824)), RIF(-RR(0.74314482547739424), -RR(0.74314482547739413))))
+            v^6 + v^5
             sage: v = QQbar(sqrt(AA(2)))
             sage: v.exactify()
             sage: sage_input(v, verify=True)
diff --git a/src/sage/rings/rational.pyx b/src/sage/rings/rational.pyx
index 272650abeef..0dbe3453dd2 100644
--- a/src/sage/rings/rational.pyx
+++ b/src/sage/rings/rational.pyx
@@ -1446,7 +1446,7 @@ cdef class Rational(sage.structure.element.FieldElement):
             sage: 0.is_norm(K)
             True
             sage: (1/7).is_norm(K, element=True)
-            (True, 1/7*beta + 3/7)
+            (True, -3/7*beta + 5/7)
             sage: (1/10).is_norm(K, element=True)
             (False, None)
             sage: (1/691).is_norm(QQ, element=True)
@@ -1558,7 +1558,7 @@ cdef class Rational(sage.structure.element.FieldElement):
         EXAMPLES::
 
             sage: QQ(2)._bnfisnorm(QuadraticField(-1, 'i'))                             # needs sage.rings.number_field
-            (i + 1, 1)
+            (-i + 1, 1)
             sage: x = polygen(QQ, 'x')
             sage: 7._bnfisnorm(NumberField(x^3 - 2, 'b'))                               # needs sage.rings.number_field
             (1, 7)
diff --git a/src/sage/schemes/affine/affine_morphism.py b/src/sage/schemes/affine/affine_morphism.py
index 0eba4c662ec..2208cefca1d 100755
--- a/src/sage/schemes/affine/affine_morphism.py
+++ b/src/sage/schemes/affine/affine_morphism.py
@@ -1158,13 +1158,11 @@ class SchemeMorphism_polynomial_affine_space_field(SchemeMorphism_polynomial_aff
             sage: H = End(A)
             sage: f = H([QQbar(3^(1/3))*x^2 + QQbar(sqrt(-2))])                         # needs sage.symbolic
             sage: f.reduce_base_field()                                                 # needs sage.symbolic
-            Scheme endomorphism of Affine Space of dimension 1 over Number
-            Field in a with defining polynomial y^6 + 6*y^4 - 6*y^3 + 12*y^2 + 36*y + 17
-             with a = 1.442249570307409? + 1.414213562373095?*I
+            Scheme endomorphism of Affine Space of dimension 1 over Number Field in a with defining polynomial y^6 + 6*y^4 - 6*y^3 + 12*y^2 + 36*y + 17 with a = 1.442249570307409? - 1.414213562373095?*I
               Defn: Defined on coordinates by sending (x) to
                     ((-48/269*a^5 + 27/269*a^4 - 320/269*a^3 + 468/269*a^2 - 772/269*a
-                    - 1092/269)*x^2 + (48/269*a^5 - 27/269*a^4 + 320/269*a^3 - 468/269*a^2
-                    + 1041/269*a + 1092/269))
+                    - 1092/269)*x^2 + (-48/269*a^5 + 27/269*a^4 - 320/269*a^3 + 468/269*a^2
+                    - 1041/269*a - 1092/269))
 
         ::
 
diff --git a/src/sage/schemes/elliptic_curves/ell_field.py b/src/sage/schemes/elliptic_curves/ell_field.py
index a63dbf57809..84a719adfbb 100755
--- a/src/sage/schemes/elliptic_curves/ell_field.py
+++ b/src/sage/schemes/elliptic_curves/ell_field.py
@@ -902,7 +902,7 @@ class EllipticCurve_field(ell_generic.EllipticCurve_generic, ProjectivePlaneCurv
              by y^2 = x^3 + 5*a0*x^2 + (-200*a0^2)*x + (-42000*a0^2+42000*a0+126000)
              over Number Field in a0 with defining polynomial x^3 - 3*x^2 + 3*x + 9
             sage: K.<b> = E.division_field(3, simplify_all=True); K
-            Number Field in b with defining polynomial x^12 - 25*x^10 + 130*x^8 + 645*x^6 + 1050*x^4 + 675*x^2 + 225
+            Number Field in b with defining polynomial x^12 + 5*x^10 + 40*x^8 + 315*x^6 + 750*x^4 + 675*x^2 + 2025
 
         Some higher-degree examples::
 
diff --git a/src/sage/schemes/elliptic_curves/ell_local_data.py b/src/sage/schemes/elliptic_curves/ell_local_data.py
index 7434659b5a2..df076ed62b6 100755
--- a/src/sage/schemes/elliptic_curves/ell_local_data.py
+++ b/src/sage/schemes/elliptic_curves/ell_local_data.py
@@ -1161,7 +1161,7 @@ def check_prime(K, P):
         sage: check_prime(K, a + 1)
         Fractional ideal (a + 1)
         sage: [check_prime(K, P) for P in K.primes_above(31)]
-        [Fractional ideal (5/2*a + 1/2), Fractional ideal (5/2*a - 1/2)]
+        [Fractional ideal (-5/2*a - 1/2), Fractional ideal (-5/2*a + 1/2)]
         sage: L.<b> = NumberField(x^2 + 3)
         sage: check_prime(K, L.ideal(5))
         Traceback (most recent call last):
diff --git a/src/sage/schemes/elliptic_curves/ell_number_field.py b/src/sage/schemes/elliptic_curves/ell_number_field.py
index 8ccdbcd452d..ffdc95bc92d 100755
--- a/src/sage/schemes/elliptic_curves/ell_number_field.py
+++ b/src/sage/schemes/elliptic_curves/ell_number_field.py
@@ -229,9 +229,9 @@ class EllipticCurve_number_field(EllipticCurve_field):
             sage: E == loads(dumps(E))
             True
             sage: E.simon_two_descent()
-            (2, 2, [(0 : 0 : 1), (1/18*a + 7/18 : -5/54*a - 17/54 : 1)])
+            (2, 2, [(0 : 0 : 1), (1/8*a + 5/8 : -3/16*a - 7/16 : 1)])
             sage: E.simon_two_descent(lim1=5, lim3=5, limtriv=10, maxprob=7, limbigprime=10)
-            (2, 2, [(-1 : 0 : 1), (-2 : -1/2*a - 1/2 : 1)])
+            (2, 2, [(-1 : 0 : 1), (1/2*a - 5/2 : -1/2*a - 13/2 : 1)])
 
         ::
 
@@ -286,7 +286,7 @@ class EllipticCurve_number_field(EllipticCurve_field):
             sage: E.simon_two_descent()  # long time (4s on sage.math, 2013)
             (3,
              3,
-             [(1/8*zeta43_0^2 - 3/8*zeta43_0 - 1/4 : -5/16*zeta43_0^2 + 7/16*zeta43_0 + 1/8 : 1),
+             [(-1/2*zeta43_0^2 - 1/2*zeta43_0 + 7 : -3/2*zeta43_0^2 - 5/2*zeta43_0 + 18 : 1),
               (0 : 0 : 1)])
         """
         verbose = int(verbose)
@@ -881,7 +881,7 @@ class EllipticCurve_number_field(EllipticCurve_field):
             sage: K.<i> = NumberField(x^2 + 1)
             sage: E = EllipticCurve([1 + i, 0, 1, 0, 0])
             sage: E.local_data()
-            [Local data at Fractional ideal (2*i + 1):
+            [Local data at Fractional ideal (-2*i - 1):
                Reduction type: bad non-split multiplicative
                Local minimal model: Elliptic Curve defined by y^2 + (i+1)*x*y + y = x^3
                                     over Number Field in i with defining polynomial x^2 + 1
@@ -889,7 +889,7 @@ class EllipticCurve_number_field(EllipticCurve_field):
                Conductor exponent: 1
                Kodaira Symbol: I1
                Tamagawa Number: 1,
-             Local data at Fractional ideal (-2*i + 3):
+             Local data at Fractional ideal (3*i + 2):
                Reduction type: bad split multiplicative
                Local minimal model: Elliptic Curve defined by y^2 + (i+1)*x*y + y = x^3
                                     over Number Field in i with defining polynomial x^2 + 1
@@ -907,7 +907,7 @@ class EllipticCurve_number_field(EllipticCurve_field):
               Kodaira Symbol: I0
               Tamagawa Number: 1
             sage: E.local_data(2*i + 1)
-            Local data at Fractional ideal (2*i + 1):
+            Local data at Fractional ideal (-2*i - 1):
               Reduction type: bad non-split multiplicative
               Local minimal model: Elliptic Curve defined by y^2 + (i+1)*x*y + y = x^3
                                    over Number Field in i with defining polynomial x^2 + 1
@@ -1465,8 +1465,10 @@ class EllipticCurve_number_field(EllipticCurve_field):
             sage: K.<a> = NumberField(x^2 - 5)
             sage: E = EllipticCurve([20, 225, 750, 625*a + 6875, 31250*a + 46875])
             sage: bad_primes = E.discriminant().support(); bad_primes
-            [Fractional ideal (-a), Fractional ideal (7/2*a - 81/2),
-             Fractional ideal (-a - 52), Fractional ideal (2)]
+            [Fractional ideal (-a),
+             Fractional ideal (-7/2*a + 81/2),
+             Fractional ideal (-a - 52),
+             Fractional ideal (2)]
             sage: [E.kodaira_symbol(P) for P in bad_primes]
             [I0, I1, I1, II]
             sage: K.<a> = QuadraticField(-11)
@@ -1492,7 +1494,7 @@ class EllipticCurve_number_field(EllipticCurve_field):
 
             sage: K.<i> = NumberField(x^2 + 1)
             sage: EllipticCurve([i, i - 1, i + 1, 24*i + 15, 14*i + 35]).conductor()
-            Fractional ideal (21*i - 3)
+            Fractional ideal (3*i + 21)
             sage: K.<a> = NumberField(x^2 - x + 3)
             sage: EllipticCurve([1 + a, -1 + a, 1 + a, -11 + a, 5 - 9*a]).conductor()
             Fractional ideal (-6*a)
@@ -2593,8 +2595,8 @@ class EllipticCurve_number_field(EllipticCurve_field):
             sage: [E1.ainvs() for E1 in C]
             [(0, 0, 0, 0, -27),
             (0, 0, 0, 0, 1),
-            (i + 1, i, i + 1, -i + 3, 4*i),
-            (i + 1, i, i + 1, -i + 33, -58*i)]
+            (i + 1, i, 0, 3, -i),
+            (i + 1, i, 0, 33, 91*i)]
 
         The matrix of degrees of cyclic isogenies between curves::
 
@@ -2625,13 +2627,13 @@ class EllipticCurve_number_field(EllipticCurve_field):
             sage: [((i,j), isogs[i][j].x_rational_map())
             ....:  for i in range(4) for j in range(4) if isogs[i][j] != 0]
             [((0, 1), (1/9*x^3 - 12)/x^2),
-             ((0, 3), (-1/2*i*x^2 + i*x - 12*i)/(x - 3)),
+             ((0, 3), (1/2*i*x^2 - 2*i*x + 15*i)/(x - 3)),
              ((1, 0), (x^3 + 4)/x^2),
-             ((1, 2), (-1/2*i*x^2 - i*x - 2*i)/(x + 1)),
-             ((2, 1), (1/2*i*x^2 - x)/(x + 3/2*i)),
-             ((2, 3), (x^3 + 4*i*x^2 - 10*x - 10*i)/(x^2 + 4*i*x - 4)),
-             ((3, 0), (1/2*i*x^2 + x + 4*i)/(x - 5/2*i)),
-             ((3, 2), (1/9*x^3 - 4/3*i*x^2 - 34/3*x + 226/9*i)/(x^2 - 8*i*x - 16))]
+             ((1, 2), (1/2*i*x^2 + i)/(x + 1)),
+             ((2, 1), (-1/2*i*x^2 - 1/2*i)/(x - 1/2*i)),
+             ((2, 3), (x^3 - 2*i*x^2 - 7*x + 4*i)/(x^2 - 2*i*x - 1)),
+             ((3, 0), (-1/2*i*x^2 + 2*x - 5/2*i)/(x + 7/2*i)),
+             ((3, 2), (1/9*x^3 + 2/3*i*x^2 - 13/3*x - 116/9*i)/(x^2 + 10*i*x - 25))]
 
         The isogeny class may be visualized by obtaining its graph and
         plotting it::
@@ -3112,10 +3114,10 @@ class EllipticCurve_number_field(EllipticCurve_field):
             sage: K.<i> = QuadraticField(-1)
             sage: E1 = EllipticCurve([i + 1, 0, 1, -240*i - 400, -2869*i - 2627])
             sage: E1.conductor()
-            Fractional ideal (-4*i - 7)
+            Fractional ideal (4*i + 7)
             sage: E2 = EllipticCurve([1+i,0,1,0,0])
             sage: E2.conductor()
-            Fractional ideal (-4*i - 7)
+            Fractional ideal (4*i + 7)
             sage: E1.is_isogenous(E2) # long time
             True
             sage: E1.is_isogenous(E2, proof=False) # faster  (~170ms)
@@ -3442,8 +3444,8 @@ class EllipticCurve_number_field(EllipticCurve_field):
             sage: Q = E(0,-1)
             sage: E.lll_reduce([P,Q])
             (
-                                                     [0 1]
-            [(0 : -1 : 1), (-2 : -1/2*a - 1/2 : 1)], [1 0]
+                                                    [ 0 -1]
+            [(0 : -1 : 1), (-2 : 1/2*a - 1/2 : 1)], [ 1  0]
             )
 
         ::
@@ -3454,9 +3456,10 @@ class EllipticCurve_number_field(EllipticCurve_field):
             ....:           E.point([-17/18*a - 1/9, -109/108*a - 277/108])]
             sage: E.lll_reduce(points)
             (
-            [(-a + 4 : -3*a + 7 : 1), (-17/18*a - 1/9 : 109/108*a + 277/108 : 1)],
-            [ 1  0]
-            [ 1 -1]
+            [(-a + 4 : -3*a + 7 : 1), (-17/18*a - 1/9 : -109/108*a - 277/108 : 1)],
+            <BLANKLINE>
+            [1 0]
+            [1 1]
             )
         """
         r = len(points)
diff --git a/src/sage/schemes/elliptic_curves/ell_point.py b/src/sage/schemes/elliptic_curves/ell_point.py
index 0da080ffdbe..34615bb46b8 100755
--- a/src/sage/schemes/elliptic_curves/ell_point.py
+++ b/src/sage/schemes/elliptic_curves/ell_point.py
@@ -142,7 +142,7 @@ lazy_import('sage.schemes.generic.morphism', 'SchemeMorphism')
 
 try:
     from sage.libs.pari.all import pari, PariError
-    from cypari2.pari_instance import prec_words_to_bits
+    from cypari2.pari_instance import prec_pari_to_bits
 except ImportError:
     PariError = ()
 
@@ -2672,10 +2672,10 @@ class EllipticCurvePoint_number_field(EllipticCurvePoint_field):
             sage: K.<i> = NumberField(x^2 + 1)
             sage: E = EllipticCurve(K, [0,1,0,-160,308])
             sage: P = E(26, -120)
-            sage: E.discriminant().support()
-            [Fractional ideal (i + 1),
-             Fractional ideal (-i - 2),
-             Fractional ideal (2*i + 1),
+            sage: E.discriminant().support() # random
+            [Fractional ideal (-i - 1),
+             Fractional ideal (2*i - 1),
+             Fractional ideal (-2*i - 1),
              Fractional ideal (3)]
             sage: [E.tamagawa_exponent(p) for p in E.discriminant().support()]
             [1, 4, 4, 4]
@@ -3705,7 +3705,7 @@ class EllipticCurvePoint_number_field(EllipticCurvePoint_field):
         E_pari = E_work.pari_curve()
         log_pari = E_pari.ellpointtoz(pt_pari, precision=working_prec)
 
-        while prec_words_to_bits(log_pari.precision()) < precision:
+        while prec_pari_to_bits(log_pari.precision()) < precision:
             # result is not precise enough, re-compute with double
             # precision. if the base field is not QQ, this
             # requires modifying the precision of the embedding,
diff --git a/src/sage/schemes/elliptic_curves/ell_rational_field.py b/src/sage/schemes/elliptic_curves/ell_rational_field.py
index 1b7a52b0ed0..466f55960f6 100755
--- a/src/sage/schemes/elliptic_curves/ell_rational_field.py
+++ b/src/sage/schemes/elliptic_curves/ell_rational_field.py
@@ -1844,7 +1844,7 @@ class EllipticCurve_rational_field(EllipticCurve_number_field):
             sage: E = EllipticCurve('389a1')
             sage: E._known_points = []  # clear cached points
             sage: E.simon_two_descent()
-            (2, 2, [(5/4 : 5/8 : 1), (-3/4 : 7/8 : 1)])
+            (2, 2, [(-3/4 : 7/8 : 1), (5/4 : 5/8 : 1)])
             sage: E = EllipticCurve('5077a1')
             sage: E.simon_two_descent()
             (3, 3, [(1 : 0 : 1), (2 : 0 : 1), (0 : 2 : 1)])
diff --git a/src/sage/schemes/elliptic_curves/gal_reps_number_field.py b/src/sage/schemes/elliptic_curves/gal_reps_number_field.py
index a4eb66a6df7..8b597e41595 100755
--- a/src/sage/schemes/elliptic_curves/gal_reps_number_field.py
+++ b/src/sage/schemes/elliptic_curves/gal_reps_number_field.py
@@ -800,17 +800,17 @@ def deg_one_primes_iter(K, principal_only=False):
         [Fractional ideal (2, a + 1),
          Fractional ideal (3, a + 1),
          Fractional ideal (3, a + 2),
-         Fractional ideal (a),
+         Fractional ideal (-a),
          Fractional ideal (7, a + 3),
          Fractional ideal (7, a + 4)]
         sage: it = deg_one_primes_iter(K, True)
         sage: [next(it) for _ in range(6)]
-        [Fractional ideal (a),
-         Fractional ideal (-2*a + 3),
-         Fractional ideal (2*a + 3),
+        [Fractional ideal (-a),
+         Fractional ideal (2*a - 3),
+         Fractional ideal (-2*a - 3),
          Fractional ideal (a + 6),
          Fractional ideal (a - 6),
-         Fractional ideal (-3*a + 4)]
+         Fractional ideal (3*a - 4)]
     """
     # imaginary quadratic fields have no principal primes of norm < disc / 4
     start = K.discriminant().abs() // 4 if principal_only and K.signature() == (0,1) else 2
diff --git a/src/sage/schemes/elliptic_curves/gp_simon.py b/src/sage/schemes/elliptic_curves/gp_simon.py
index 729380dccf1..d18f753c76c 100644
--- a/src/sage/schemes/elliptic_curves/gp_simon.py
+++ b/src/sage/schemes/elliptic_curves/gp_simon.py
@@ -59,7 +59,7 @@ def simon_two_descent(E, verbose=0, lim1=None, lim3=None, limtriv=None,
         sage: import sage.schemes.elliptic_curves.gp_simon
         sage: E = EllipticCurve('389a1')
         sage: sage.schemes.elliptic_curves.gp_simon.simon_two_descent(E)
-        (2, 2, [(5/4 : 5/8 : 1), (-3/4 : 7/8 : 1)])
+        (2, 2, [(-3/4 : 7/8 : 1), (5/4 : 5/8 : 1)])
 
     TESTS::
 
diff --git a/src/sage/schemes/elliptic_curves/isogeny_class.py b/src/sage/schemes/elliptic_curves/isogeny_class.py
index 13edc68a022..a83fd8705ac 100755
--- a/src/sage/schemes/elliptic_curves/isogeny_class.py
+++ b/src/sage/schemes/elliptic_curves/isogeny_class.py
@@ -223,8 +223,8 @@ class IsogenyClass_EC(SageObject):
             sage: C.curves
             [Elliptic Curve defined by y^2 = x^3 + (-27) over Number Field in i with defining polynomial x^2 + 1 with i = 1*I,
              Elliptic Curve defined by y^2 = x^3 + 1 over Number Field in i with defining polynomial x^2 + 1 with i = 1*I,
-             Elliptic Curve defined by y^2 + (i+1)*x*y + (i+1)*y = x^3 + i*x^2 + (-i+3)*x + 4*i over Number Field in i with defining polynomial x^2 + 1 with i = 1*I,
-             Elliptic Curve defined by y^2 + (i+1)*x*y + (i+1)*y = x^3 + i*x^2 + (-i+33)*x + (-58*i) over Number Field in i with defining polynomial x^2 + 1 with i = 1*I]
+             Elliptic Curve defined by y^2 + (i+1)*x*y = x^3 + i*x^2 + 3*x + (-i) over Number Field in i with defining polynomial x^2 + 1 with i = 1*I,
+             Elliptic Curve defined by y^2 + (i+1)*x*y = x^3 + i*x^2 + 33*x + 91*i over Number Field in i with defining polynomial x^2 + 1 with i = 1*I]
         """
         if self._label:
             return "Elliptic curve isogeny class %s" % (self._label)
@@ -615,8 +615,8 @@ class IsogenyClass_EC_NumberField(IsogenyClass_EC):
             sage: [E1.ainvs() for E1 in C]
             [(0, 0, 0, 0, -27),
              (0, 0, 0, 0, 1),
-             (i + 1, i, i + 1, -i + 3, 4*i),
-             (i + 1, i, i + 1, -i + 33, -58*i)]
+             (i + 1, i, 0, 3, -i),
+             (i + 1, i, 0, 33, 91*i)]
 
         The matrix of degrees of cyclic isogenies between curves::
 
@@ -647,13 +647,13 @@ class IsogenyClass_EC_NumberField(IsogenyClass_EC):
             sage: [((i,j), isogs[i][j].x_rational_map())
             ....:  for i in range(4) for j in range(4) if isogs[i][j] != 0]
             [((0, 1), (1/9*x^3 - 12)/x^2),
-             ((0, 3), (-1/2*i*x^2 + i*x - 12*i)/(x - 3)),
+             ((0, 3), (1/2*i*x^2 - 2*i*x + 15*i)/(x - 3)),
              ((1, 0), (x^3 + 4)/x^2),
-             ((1, 2), (-1/2*i*x^2 - i*x - 2*i)/(x + 1)),
-             ((2, 1), (1/2*i*x^2 - x)/(x + 3/2*i)),
-             ((2, 3), (x^3 + 4*i*x^2 - 10*x - 10*i)/(x^2 + 4*i*x - 4)),
-             ((3, 0), (1/2*i*x^2 + x + 4*i)/(x - 5/2*i)),
-             ((3, 2), (1/9*x^3 - 4/3*i*x^2 - 34/3*x + 226/9*i)/(x^2 - 8*i*x - 16))]
+             ((1, 2), (1/2*i*x^2 + i)/(x + 1)),
+             ((2, 1), (-1/2*i*x^2 - 1/2*i)/(x - 1/2*i)),
+             ((2, 3), (x^3 - 2*i*x^2 - 7*x + 4*i)/(x^2 - 2*i*x - 1)),
+             ((3, 0), (-1/2*i*x^2 + 2*x - 5/2*i)/(x + 7/2*i)),
+             ((3, 2), (1/9*x^3 + 2/3*i*x^2 - 13/3*x - 116/9*i)/(x^2 + 10*i*x - 25))]
 
             sage: K.<i> = QuadraticField(-1)
             sage: E = EllipticCurve([1+i, -i, i, 1, 0])
diff --git a/src/sage/schemes/elliptic_curves/isogeny_small_degree.py b/src/sage/schemes/elliptic_curves/isogeny_small_degree.py
index 6a0194fb0f9..90f7382a94e 100755
--- a/src/sage/schemes/elliptic_curves/isogeny_small_degree.py
+++ b/src/sage/schemes/elliptic_curves/isogeny_small_degree.py
@@ -886,15 +886,15 @@ def isogenies_5_0(E, minimal_models=True):
           from Elliptic Curve defined by y^2 + y = x^3
                over Number Field in a with defining polynomial x^6 - 320*x^3 - 320
             to Elliptic Curve defined by
-               y^2 + y = x^3 + (241565/32*a^5-362149/48*a^4+180281/24*a^3-9693307/4*a^2+14524871/6*a-7254985/3)*x
-                          + (1660391123/192*a^5-829315373/96*a^4+77680504/9*a^3-66622345345/24*a^2+33276655441/12*a-24931615912/9)
+               y^2 + y = x^3 + (643/8*a^5-15779/48*a^4-32939/24*a^3-71989/2*a^2+214321/6*a-112115/3)*x
+                          + (2901961/96*a^5+4045805/48*a^4+12594215/18*a^3-30029635/6*a^2+15341626/3*a-38944312/9)
                over Number Field in a with defining polynomial x^6 - 320*x^3 - 320,
          Isogeny of degree 5
           from Elliptic Curve defined by y^2 + y = x^3
                over Number Field in a with defining polynomial x^6 - 320*x^3 - 320
             to Elliptic Curve defined by
-               y^2 + y = x^3 + (47519/32*a^5-72103/48*a^4+32939/24*a^3-1909753/4*a^2+2861549/6*a-1429675/3)*x
-                          + (-131678717/192*a^5+65520419/96*a^4-12594215/18*a^3+5280985135/24*a^2-2637787519/12*a+1976130088/9)
+               y^2 + y = x^3 + (-1109/8*a^5-53873/48*a^4-180281/24*a^3-14491/2*a^2+35899/6*a-43745/3)*x
+                          + (-17790679/96*a^5-60439571/48*a^4-77680504/9*a^3+1286245/6*a^2-4961854/3*a-73854632/9)
                over Number Field in a with defining polynomial x^6 - 320*x^3 - 320]
     """
     F = E.base_field()
diff --git a/src/sage/schemes/plane_conics/con_number_field.py b/src/sage/schemes/plane_conics/con_number_field.py
index e09a1f60262..69fe9960c3d 100755
--- a/src/sage/schemes/plane_conics/con_number_field.py
+++ b/src/sage/schemes/plane_conics/con_number_field.py
@@ -121,7 +121,7 @@ class ProjectiveConic_number_field(ProjectiveConic_field):
             sage: K.<i> = QuadraticField(-1)
             sage: C = Conic(K, [1, 3, -5])
             sage: C.has_rational_point(point=True, obstruction=True)
-            (False, Fractional ideal (-i - 2))
+            (False, Fractional ideal (i + 2))
             sage: C.has_rational_point(algorithm='rnfisnorm')
             False
             sage: C.has_rational_point(algorithm='rnfisnorm', obstruction=True,
@@ -135,7 +135,7 @@ class ProjectiveConic_number_field(ProjectiveConic_field):
             sage: L.<b> = NumberField(x^3 - 5)
             sage: C = Conic(L, [1, 2, -3])
             sage: C.has_rational_point(point=True, algorithm='rnfisnorm')
-            (True, (5/3 : -1/3 : 1))
+            (True, (-5/3 : 1/3 : 1))
 
             sage: K.<a> = NumberField(x^4+2)
             sage: Conic(QQ, [4,5,6]).has_rational_point()
diff --git a/src/sage/schemes/projective/projective_morphism.py b/src/sage/schemes/projective/projective_morphism.py
index 20031e81a41..fcbb0c01e82 100755
--- a/src/sage/schemes/projective/projective_morphism.py
+++ b/src/sage/schemes/projective/projective_morphism.py
@@ -928,7 +928,7 @@ class SchemeMorphism_polynomial_projective_space(SchemeMorphism_polynomial):
             Dynamical System of Projective Space of dimension 1 over
             Number Field in a with defining polynomial 3*x^2 + 1
             Defn: Defined on coordinates by sending (z : w) to
-                ((-3/2*a + 1/2)*z^2 + (-3/2*a + 1/2)*w^2 : (-3/2*a - 3/2)*z*w)
+                ((3/2*a + 1/2)*z^2 + (3/2*a + 1/2)*w^2 : (-3/2*a + 3/2)*z*w)
 
         ::
 
@@ -1728,11 +1728,11 @@ class SchemeMorphism_polynomial_projective_space_field(SchemeMorphism_polynomial
             sage: f._number_field_from_algebraics()                                     # needs sage.symbolic
             Scheme endomorphism of Projective Space of dimension 1 over Number
              Field in a with defining polynomial y^6 + 6*y^4 - 6*y^3 + 12*y^2 + 36*y + 17
-             with a = 1.442249570307409? + 1.414213562373095?*I
+             with a = 1.442249570307409? - 1.414213562373095?*I
               Defn: Defined on coordinates by sending (x : y) to
                     ((-48/269*a^5 + 27/269*a^4 - 320/269*a^3 + 468/269*a^2 - 772/269*a
-                    - 1092/269)*x^2 + (48/269*a^5 - 27/269*a^4 + 320/269*a^3 - 468/269*a^2
-                    + 1041/269*a + 1092/269)*y^2 : y^2)
+                    - 1092/269)*x^2 + (-48/269*a^5 + 27/269*a^4 - 320/269*a^3 + 468/269*a^2
+                    - 1041/269*a - 1092/269)*y^2 : y^2)
 
         ::
 
@@ -1745,12 +1745,12 @@ class SchemeMorphism_polynomial_projective_space_field(SchemeMorphism_polynomial
             Scheme morphism:
               From: Projective Space of dimension 1 over Number Field in a
                     with defining polynomial y^4 + 3*y^2 + 1
-                    with a = 0.?e-113 + 0.618033988749895?*I
+                    with a = 0.?e-166 + 1.618033988749895?*I
               To:   Projective Space of dimension 2 over Number Field in a
                     with defining polynomial y^4 + 3*y^2 + 1
-                    with a = 0.?e-113 + 0.618033988749895?*I
+                    with a = 0.?e-166 + 1.618033988749895?*I
               Defn: Defined on coordinates by sending (x : y) to
-                    (x^2 + (a^3 + 2*a)*x*y + 3*y^2 : y^2 : (2*a^2 + 3)*x*y)
+                    (x^2 + (-a^3 - 2*a)*x*y + 3*y^2 : y^2 : (-2*a^2 - 3)*x*y)
 
         The following was fixed in :issue:`23808`::
 
diff --git a/src/sage/schemes/projective/projective_point.py b/src/sage/schemes/projective/projective_point.py
index 7f941ec6726..b0f443acb4e 100755
--- a/src/sage/schemes/projective/projective_point.py
+++ b/src/sage/schemes/projective/projective_point.py
@@ -1238,10 +1238,10 @@ class SchemeMorphism_point_projective_field(SchemeMorphism_point_projective_ring
             sage: P.<x,y> = ProjectiveSpace(QQbar, 1)
             sage: Q = P([-1/2*QQbar(sqrt(2)) + QQbar(I), 1])
             sage: S = Q._number_field_from_algebraics(); S
-            (1/2*a^3 + a^2 - 1/2*a : 1)
+            (-1/2*a^3 + a^2 + 1/2*a : 1)
             sage: S.codomain()
             Projective Space of dimension 1 over Number Field in a with defining
-             polynomial y^4 + 1 with a = 0.7071067811865475? + 0.7071067811865475?*I
+             polynomial y^4 + 1 with a = -0.7071067811865475? - 0.7071067811865475?*I
 
         The following was fixed in :issue:`23808`::
 
@@ -1251,7 +1251,7 @@ class SchemeMorphism_point_projective_field(SchemeMorphism_point_projective_ring
             sage: Q = P([-1/2*QQbar(sqrt(2)) + QQbar(I), 1]);Q
             (-0.7071067811865475? + 1*I : 1)
             sage: S = Q._number_field_from_algebraics(); S
-            (1/2*a^3 + a^2 - 1/2*a : 1)
+            (-1/2*a^3 + a^2 + 1/2*a : 1)
             sage: T = S.change_ring(QQbar)  # Used to fail
             sage: T
             (-0.7071067811865475? + 1.000000000000000?*I : 1)
diff --git a/src/sage/structure/factorization.py b/src/sage/structure/factorization.py
index ab3fa717031..b16822791dc 100644
--- a/src/sage/structure/factorization.py
+++ b/src/sage/structure/factorization.py
@@ -143,17 +143,17 @@ Factorizations can involve fairly abstract mathematical objects::
     sage: K.<a> = NumberField(x^2 + 3); K
     Number Field in a with defining polynomial x^2 + 3
     sage: f = K.factor(15); f
-    (Fractional ideal (1/2*a + 3/2))^2 * (Fractional ideal (5))
+    (Fractional ideal (-a))^2 * (Fractional ideal (5))
     sage: f.universe()
     Monoid of ideals of Number Field in a with defining polynomial x^2 + 3
     sage: f.unit()
     Fractional ideal (1)
     sage: g = K.factor(9); g
-    (Fractional ideal (1/2*a + 3/2))^4
+    (Fractional ideal (-a))^4
     sage: f.lcm(g)
-    (Fractional ideal (1/2*a + 3/2))^4 * (Fractional ideal (5))
+    (Fractional ideal (-a))^4 * (Fractional ideal (5))
     sage: f.gcd(g)
-    (Fractional ideal (1/2*a + 3/2))^2
+    (Fractional ideal (-a))^2
     sage: f.is_integral()
     True