diff --git a/src/sage/libs/eclib/__init__.pxd b/src/sage/libs/eclib/__init__.pxd
index 3f99f99..d44d4fb 100644
--- a/src/sage/libs/eclib/__init__.pxd
+++ b/src/sage/libs/eclib/__init__.pxd
@@ -12,9 +12,11 @@ from libcpp.pair cimport pair
 from sage.libs.ntl.types cimport ZZ_c
 
 
-# NOTE: eclib includes have specific dependencies and must be included
-# in a specific order. So we start by listing all relevant include files
-# in the correct order.
+# NOTE: eclib used to have specific dependencies, so that they had to
+# be included in a specific order. Although this is no longer the
+# case, we start by listing all relevant include files in the correct
+# order.
+
 cdef extern from "eclib/vector.h": pass
 cdef extern from "eclib/xmod.h": pass
 cdef extern from "eclib/svector.h": pass
diff --git a/src/sage/libs/eclib/interface.py b/src/sage/libs/eclib/interface.py
index e898456..493b5f1 100644
--- a/src/sage/libs/eclib/interface.py
+++ b/src/sage/libs/eclib/interface.py
@@ -21,17 +21,16 @@ Check that ``eclib`` is imported as needed::
     sage: [k for k in sys.modules if k.startswith("sage.libs.eclib")]
     []
     sage: EllipticCurve('11a1').mwrank_curve()
-    y^2+ y = x^3 - x^2 - 10*x - 20
+    y^2 + y = x^3 - x^2 - 10 x - 20
     sage: [k for k in sys.modules if k.startswith("sage.libs.eclib")]
     ['...']
 """
-
+import sys
 from sage.structure.sage_object import SageObject
 from sage.rings.all import Integer
 from sage.rings.integer_ring import IntegerRing
 
-from .mwrank import _Curvedata, _two_descent, _mw
-
+from .mwrank import _Curvedata, _two_descent, _mw, parse_point_list
 
 class mwrank_EllipticCurve(SageObject):
     r"""
@@ -67,7 +66,7 @@ class mwrank_EllipticCurve(SageObject):
 
         sage: e = mwrank_EllipticCurve([3, -4])
         sage: e
-        y^2 = x^3 + 3*x - 4
+        y^2 = x^3 + 3 x - 4
         sage: e.ainvs()
         [0, 0, 0, 3, -4]
 
@@ -127,6 +126,7 @@ class mwrank_EllipticCurve(SageObject):
 
         # place holders
         self.__saturate = -2  # not yet saturated
+        self.__descent = None
 
     def __reduce__(self):
         r"""
@@ -137,12 +137,9 @@ class mwrank_EllipticCurve(SageObject):
             sage: E = mwrank_EllipticCurve([0,0,1,-7,6])
             sage: E.__reduce__()
             (<class 'sage.libs.eclib.interface.mwrank_EllipticCurve'>, ([0, 0, 1, -7, 6], False))
-
-
         """
         return mwrank_EllipticCurve, (self.__ainvs, self.__verbose)
 
-
     def set_verbose(self, verbose):
         """
         Set the verbosity of printing of output by the :meth:`two_descent()` and
@@ -247,53 +244,27 @@ class mwrank_EllipticCurve(SageObject):
 
             sage: E = mwrank_EllipticCurve([0,-1,1,0,0])
             sage: E.__repr__()
-            'y^2+ y = x^3 - x^2 '
+            'y^2 + y = x^3 - x^2'
         """
-        # TODO: Is the use (or omission) of spaces here intentional?
-        a = self.ainvs()
-        s = "y^2"
-        if a[0] == -1:
-            s += "- x*y "
-        elif a[0] == 1:
-            s += "+ x*y "
-        elif a[0] != 0:
-            s += "+ %s*x*y "%a[0]
-        if a[2] == -1:
-            s += " - y"
-        elif a[2] == 1:
-            s += "+ y"
-        elif a[2] != 0:
-            s += "+ %s*y"%a[2]
-        s += " = x^3 "
-        if a[1] == -1:
-            s += "- x^2 "
-        elif a[1] == 1:
-            s += "+ x^2 "
-        elif a[1] != 0:
-            s += "+ %s*x^2 "%a[1]
-        if a[3] == -1:
-            s += "- x "
-        elif a[3] == 1:
-            s += "+ x "
-        elif a[3] != 0:
-            s += "+ %s*x "%a[3]
-        if a[4] == -1:
-            s += "-1"
-        elif a[4] == 1:
-            s += "+1"
-        elif a[4] != 0:
-            s += "+ %s"%a[4]
-        s = s.replace("+ -","- ")
-        return s
-
+        a1, a2, a3, a4, a6 = self.__ainvs
+        # we do not assume a1, a2, a3 are reduced to {0,1}, {-1,0,1}, {0,1}
+        coeff = lambda a: ''.join([" +" if a > 0 else " -",
+                                   " " + str(abs(a)) if abs(a) > 1 else ""])
+        return ''.join(['y^2',
+                        ' '.join([coeff(a1), 'xy']) if a1 else '',
+                        ' '.join([coeff(a3), 'y']) if a3 else '',
+                        ' = x^3',
+                        ' '.join([coeff(a2), 'x^2']) if a2 else '',
+                        ' '.join([coeff(a4), 'x']) if a4 else '',
+                        ' '.join([" +" if a6 > 0 else " -", str(abs(a6))]) if a6 else ''])
 
     def two_descent(self,
-                    verbose = True,
-                    selmer_only = False,
-                    first_limit = 20,
-                    second_limit = 8,
-                    n_aux = -1,
-                    second_descent = True):
+                    verbose=True,
+                    selmer_only=False,
+                    first_limit=20,
+                    second_limit=8,
+                    n_aux=-1,
+                    second_descent=True):
         r"""
         Compute 2-descent data for this curve.
 
@@ -374,16 +345,14 @@ class mwrank_EllipticCurve(SageObject):
         second_limit = int(second_limit)
         n_aux = int(n_aux)
         second_descent = int(second_descent)    # convert from bool to (int) 0 or 1
-        # TODO:  Don't allow limits above some value...???
-        #   (since otherwise mwrank just sets limit tiny)
         self.__descent = _two_descent()
         self.__descent.do_descent(self.__curve,
-                                      verbose,
-                                      selmer_only,
-                                      first_limit,
-                                      second_limit,
-                                      n_aux,
-                                      second_descent)
+                                  verbose,
+                                  selmer_only,
+                                  first_limit,
+                                  second_limit,
+                                  n_aux,
+                                  second_descent)
         if not self.__descent.ok():
             raise RuntimeError("A 2-descent did not complete successfully.")
         self.__saturate = -2  # not yet saturated
@@ -398,11 +367,9 @@ class mwrank_EllipticCurve(SageObject):
             sage: E._mwrank_EllipticCurve__two_descent_data()
             <sage.libs.eclib.mwrank._two_descent object at ...>
         """
-        try:
-            return self.__descent
-        except AttributeError:
+        if self.__descent is None:
             self.two_descent(self.__verbose)
-            return self.__descent
+        return self.__descent
 
     def conductor(self):
         """
@@ -565,22 +532,24 @@ class mwrank_EllipticCurve(SageObject):
         R = self.__two_descent_data().regulator()
         return float(R)
 
-    def saturate(self, bound=-1):
+    def saturate(self, bound=-1, lower=2):
         """
-        Compute the saturation of the Mordell-Weil group at all
-        primes up to ``bound``.
+        Compute the saturation of the Mordell-Weil group.
 
         INPUT:
 
-        - ``bound`` (int, default -1) -- Use `-1` (the default) to
-          saturate at *all* primes, `0` for no saturation, or `n` (a
-          positive integer) to saturate at all primes up to `n`.
+        - ``bound`` (int, default -1) -- If `-1`, saturate at *all*
+          primes by computing a bound on the saturation index,
+          otherwise saturate at all primes up to the minimum of
+          ``bound`` and the saturation index bound.
+
+        - ``lower`` (int, default 2) -- Only saturate at primes not
+          less than this.
 
         EXAMPLES:
 
         Since the 2-descent automatically saturates at primes up to
-        20, it is not easy to come up with an example where saturation
-        has any effect::
+        20, further saturation often has no effect::
 
             sage: E = mwrank_EllipticCurve([0, 0, 0, -1002231243161, 0])
             sage: E.gens()
@@ -599,7 +568,7 @@ class mwrank_EllipticCurve(SageObject):
         """
         bound = int(bound)
         if self.__saturate < bound:
-            self.__two_descent_data().saturate(bound)
+            self.__two_descent_data().saturate(bound, lower)
             self.__saturate = bound
 
     def gens(self):
@@ -613,8 +582,7 @@ class mwrank_EllipticCurve(SageObject):
             [[0, -1, 1]]
         """
         self.saturate()
-        L = eval(self.__two_descent_data().getbasis().replace(":",","))
-        return [[Integer(x), Integer(y), Integer(z)] for (x,y,z) in L]
+        return parse_point_list(self.__two_descent_data().getbasis())
 
     def certain(self):
         r"""
@@ -760,65 +728,37 @@ class mwrank_MordellWeil(SageObject):
         sage: EQ.search(1)
         P1 = [0:1:0]     is torsion point, order 1
         P1 = [-3:0:1]     is generator number 1
-        saturating up to 20...Checking 2-saturation
-        Points have successfully been 2-saturated (max q used = 7)
-        Checking 3-saturation
-        Points have successfully been 3-saturated (max q used = 7)
-        Checking 5-saturation
-        Points have successfully been 5-saturated (max q used = 23)
-        Checking 7-saturation
-        Points have successfully been 7-saturated (max q used = 41)
-        Checking 11-saturation
-        Points have successfully been 11-saturated (max q used = 17)
-        Checking 13-saturation
-        Points have successfully been 13-saturated (max q used = 43)
-        Checking 17-saturation
-        Points have successfully been 17-saturated (max q used = 31)
-        Checking 19-saturation
-        Points have successfully been 19-saturated (max q used = 37)
+        saturating up to 20...Saturation index bound (for points of good reduction)  = 3
+        Reducing saturation bound from given value 20 to computed index bound 3
+        Checking saturation at [ 2 3 ]
+        Checking 2-saturation 
+        Points were proved 2-saturated (max q used = 7)
+        Checking 3-saturation 
+        Points were proved 3-saturated (max q used = 7)
         done
         P2 = [-2:3:1]     is generator number 2
-        saturating up to 20...Checking 2-saturation
+        saturating up to 20...Saturation index bound (for points of good reduction)  = 4
+        Reducing saturation bound from given value 20 to computed index bound 4
+        Checking saturation at [ 2 3 ]
+        Checking 2-saturation 
         possible kernel vector = [1,1]
         This point may be in 2E(Q): [14:-52:1]
-        ...and it is!
+        ...and it is! 
         Replacing old generator #1 with new generator [1:-1:1]
+        Reducing index bound from 4 to 2
         Points have successfully been 2-saturated (max q used = 7)
         Index gain = 2^1
-        Checking 3-saturation
-        Points have successfully been 3-saturated (max q used = 13)
-        Checking 5-saturation
-        Points have successfully been 5-saturated (max q used = 67)
-        Checking 7-saturation
-        Points have successfully been 7-saturated (max q used = 53)
-        Checking 11-saturation
-        Points have successfully been 11-saturated (max q used = 73)
-        Checking 13-saturation
-        Points have successfully been 13-saturated (max q used = 103)
-        Checking 17-saturation
-        Points have successfully been 17-saturated (max q used = 113)
-        Checking 19-saturation
-        Points have successfully been 19-saturated (max q used = 47)
-        done (index = 2).
+        done, index = 2.
         Gained index 2, new generators = [ [1:-1:1] [-2:3:1] ]
         P3 = [-14:25:8]   is generator number 3
-        saturating up to 20...Checking 2-saturation
-        Points have successfully been 2-saturated (max q used = 11)
-        Checking 3-saturation
-        Points have successfully been 3-saturated (max q used = 13)
-        Checking 5-saturation
-        Points have successfully been 5-saturated (max q used = 71)
-        Checking 7-saturation
-        Points have successfully been 7-saturated (max q used = 101)
-        Checking 11-saturation
-        Points have successfully been 11-saturated (max q used = 127)
-        Checking 13-saturation
-        Points have successfully been 13-saturated (max q used = 151)
-        Checking 17-saturation
-        Points have successfully been 17-saturated (max q used = 139)
-        Checking 19-saturation
-        Points have successfully been 19-saturated (max q used = 179)
-        done (index = 1).
+        saturating up to 20...Saturation index bound (for points of good reduction)  = 3
+        Reducing saturation bound from given value 20 to computed index bound 3
+        Checking saturation at [ 2 3 ]
+        Checking 2-saturation 
+        Points were proved 2-saturated (max q used = 11)
+        Checking 3-saturation 
+        Points were proved 3-saturated (max q used = 13)
+        done, index = 1.
         P4 = [-1:3:1]    = -1*P1 + -1*P2 + -1*P3 (mod torsion)
         P4 = [0:2:1]     = 2*P1 + 0*P2 + 1*P3 (mod torsion)
         P4 = [2:13:8]    = -3*P1 + 1*P2 + -1*P3 (mod torsion)
@@ -878,7 +818,7 @@ class mwrank_MordellWeil(SageObject):
             sage: E = mwrank_EllipticCurve([0,0,1,-7,6])
             sage: EQ = mwrank_MordellWeil(E)
             sage: EQ.__reduce__()
-            (<class 'sage.libs.eclib.interface.mwrank_MordellWeil'>, (y^2+ y = x^3 - 7*x + 6, True, 1, 999))
+            (<class 'sage.libs.eclib.interface.mwrank_MordellWeil'>, (y^2 + y = x^3 - 7 x + 6, True, 1, 999))
         """
         return mwrank_MordellWeil, (self.__curve, self.__verbose, self.__pp, self.__maxr)
 
@@ -902,12 +842,10 @@ class mwrank_MordellWeil(SageObject):
         """
         return "Subgroup of Mordell-Weil group: %s"%self.__mw
 
-    def process(self, v, sat=0):
-        """
-        This function allows one to add points to a :class:`mwrank_MordellWeil` object.
+    def process(self, v, saturation_bound=0):
+        """Process points in the list ``v``.
 
-        Process points in the list ``v``, with saturation at primes up to
-        ``sat``.  If ``sat`` is zero (the default), do no saturation.
+        This function allows one to add points to a :class:`mwrank_MordellWeil` object.
 
         INPUT:
 
@@ -915,8 +853,9 @@ class mwrank_MordellWeil(SageObject):
           list of triples of integers, which define points on the
           curve.
 
-        - ``sat`` (int, default 0) -- saturate at primes up to ``sat``, or at
-          *all* primes if ``sat`` is zero.
+        - ``saturation_bound`` (int, default 0) -- saturate at primes up to
+          ``saturation_bound``, or at *all* primes if ``saturation_bound`` is -1; when ``saturation_bound``
+          is 0 (the default), do no saturation..
 
         OUTPUT:
 
@@ -939,11 +878,11 @@ class mwrank_MordellWeil(SageObject):
             sage: EQ.points()
             [[1, -1, 1], [-2, 3, 1], [-14, 25, 8]]
 
-        Example to illustrate the saturation parameter ``sat``::
+        Example to illustrate the saturation parameter ``saturation_bound``::
 
             sage: E = mwrank_EllipticCurve([0,0,1,-7,6])
             sage: EQ = mwrank_MordellWeil(E)
-            sage: EQ.process([[1547, -2967, 343], [2707496766203306, 864581029138191, 2969715140223272], [-13422227300, -49322830557, 12167000000]], sat=20)
+            sage: EQ.process([[1547, -2967, 343], [2707496766203306, 864581029138191, 2969715140223272], [-13422227300, -49322830557, 12167000000]], saturation_bound=20)
             P1 = [1547:-2967:343]         is generator number 1
             ...
             Gained index 5, new generators = [ [-2:3:1] [-14:25:8] [1:-1:1] ]
@@ -956,7 +895,7 @@ class mwrank_MordellWeil(SageObject):
 
             sage: E = mwrank_EllipticCurve([0,0,1,-7,6])
             sage: EQ = mwrank_MordellWeil(E)
-            sage: EQ.process([[1547, -2967, 343], [2707496766203306, 864581029138191, 2969715140223272], [-13422227300, -49322830557, 12167000000]], sat=0)
+            sage: EQ.process([[1547, -2967, 343], [2707496766203306, 864581029138191, 2969715140223272], [-13422227300, -49322830557, 12167000000]], saturation_bound=0)
             P1 = [1547:-2967:343]         is generator number 1
             P2 = [2707496766203306:864581029138191:2969715140223272]      is generator number 2
             P3 = [-13422227300:-49322830557:12167000000]          is generator number 3
@@ -965,55 +904,92 @@ class mwrank_MordellWeil(SageObject):
             sage: EQ.regulator()
             375.42920288254555
             sage: EQ.saturate(2)  # points were not 2-saturated
-            saturating basis...Saturation index bound = 93
-            WARNING: saturation at primes p > 2 will not be done;
-            ...
+            saturating basis...Saturation index bound (for points of good reduction)  = 93
+            Only p-saturating for p up to given value 2.
+            The resulting points may not be p-saturated for p between this and the computed index bound 93
+            Checking saturation at [ 2 ]
+            Checking 2-saturation 
+            possible kernel vector = [1,0,0]
+            This point may be in 2E(Q): [1547:-2967:343]
+            ...and it is! 
+            Replacing old generator #1 with new generator [-2:3:1]
+            Reducing index bound from 93 to 46
+            Points have successfully been 2-saturated (max q used = 11)
+            Index gain = 2^1
+            done
             Gained index 2
-            New regulator =  93.857...
-            (False, 2, '[ ]')
+            New regulator =  93.85730072
+            (True, 2, '[ ]')
             sage: EQ.points()
             [[-2, 3, 1], [2707496766203306, 864581029138191, 2969715140223272], [-13422227300, -49322830557, 12167000000]]
             sage: EQ.regulator()
             93.85730072063639
             sage: EQ.saturate(3)  # points were not 3-saturated
-            saturating basis...Saturation index bound = 46
-            WARNING: saturation at primes p > 3 will not be done;
-            ...
+            saturating basis...Saturation index bound (for points of good reduction)  = 46
+            Only p-saturating for p up to given value 3.
+            The resulting points may not be p-saturated for p between this and the computed index bound 46
+            Checking saturation at [ 2 3 ]
+            Checking 2-saturation 
+            Points were proved 2-saturated (max q used = 11)
+            Checking 3-saturation 
+            possible kernel vector = [0,1,0]
+            This point may be in 3E(Q): [2707496766203306:864581029138191:2969715140223272]
+            ...and it is! 
+            Replacing old generator #2 with new generator [-14:25:8]
+            Reducing index bound from 46 to 15
+            Points have successfully been 3-saturated (max q used = 13)
+            Index gain = 3^1
+            done
             Gained index 3
-            New regulator =  10.428...
-            (False, 3, '[ ]')
+            New regulator =  10.42858897
+            (True, 3, '[ ]')
             sage: EQ.points()
             [[-2, 3, 1], [-14, 25, 8], [-13422227300, -49322830557, 12167000000]]
             sage: EQ.regulator()
             10.4285889689596
             sage: EQ.saturate(5)  # points were not 5-saturated
-            saturating basis...Saturation index bound = 15
-            WARNING: saturation at primes p > 5 will not be done;
-            ...
+            saturating basis...Saturation index bound (for points of good reduction)  = 15
+            Only p-saturating for p up to given value 5.
+            The resulting points may not be p-saturated for p between this and the computed index bound 15
+            Checking saturation at [ 2 3 5 ]
+            Checking 2-saturation 
+            Points were proved 2-saturated (max q used = 11)
+            Checking 3-saturation 
+            Points were proved 3-saturated (max q used = 13)
+            Checking 5-saturation 
+            possible kernel vector = [0,0,1]
+            This point may be in 5E(Q): [-13422227300:-49322830557:12167000000]
+            ...and it is! 
+            Replacing old generator #3 with new generator [1:-1:1]
+            Reducing index bound from 15 to 3
+            Points have successfully been 5-saturated (max q used = 71)
+            Index gain = 5^1
+            done
             Gained index 5
-            New regulator =  0.417...
-            (False, 5, '[ ]')
+            New regulator =  0.4171435588
+            (True, 5, '[ ]')
             sage: EQ.points()
             [[-2, 3, 1], [-14, 25, 8], [1, -1, 1]]
             sage: EQ.regulator()
             0.417143558758384
             sage: EQ.saturate()   # points are now saturated
-            saturating basis...Saturation index bound = 3
+            saturating basis...Saturation index bound (for points of good reduction)  = 3
+            Tamagawa index primes are [ ]
             Checking saturation at [ 2 3 ]
-            Checking 2-saturation
+            Checking 2-saturation 
             Points were proved 2-saturated (max q used = 11)
-            Checking 3-saturation
+            Checking 3-saturation 
             Points were proved 3-saturated (max q used = 13)
             done
             (True, 1, '[ ]')
         """
         if not isinstance(v, list):
             raise TypeError("v (=%s) must be a list"%v)
-        sat = int(sat)
+        saturation_bound = int(saturation_bound)
         for P in v:
-            if not isinstance(P, (list,tuple)) or len(P) != 3:
+            if not isinstance(P, (list, tuple)) or len(P) != 3:
                 raise TypeError("v (=%s) must be a list of 3-tuples (or 3-element lists) of ints"%v)
-            self.__mw.process(P, sat)
+            self.__mw.process(P, saturation_bound)
 
     def regulator(self):
         """
@@ -1091,23 +1067,21 @@ class mwrank_MordellWeil(SageObject):
         """
         return self.__mw.rank()
 
-    def saturate(self, max_prime=-1, odd_primes_only=False):
-        r"""
-        Saturate this subgroup of the Mordell-Weil group.
+    def saturate(self, max_prime=-1, min_prime=2):
+        r"""Saturate this subgroup of the Mordell-Weil group.
 
         INPUT:
 
-        - ``max_prime`` (int, default -1) -- saturation is performed for
-          all primes up to ``max_prime``. If `-1` (the default), an
+        - ``max_prime`` (int, default -1) -- If `-1` (the default), an
           upper bound is computed for the primes at which the subgroup
-          may not be saturated, and this is used; however, if the
-          computed bound is greater than a value set by the ``eclib``
-          library (currently 97) then no saturation will be attempted
-          at primes above this.
+          may not be saturated, and saturation is performed for all
+          primes up to this bound.  Otherwise, the bound used is the
+          minimum of ``max_prime`` and the computed bound.
 
-        - ``odd_primes_only`` (bool, default ``False``) -- only do
-          saturation at odd primes.  (If the points have been found
-          via :meth:`two_descent` they should already be 2-saturated.)
+        - ``min_prime`` (int, default 2) -- only do saturation at
+          primes no less than this.  (For example, if the points have
+          been found via :meth:`two_descent` they should already be
+          2-saturated so a value of 3 is appropriate.)
 
         OUTPUT:
 
@@ -1115,40 +1089,35 @@ class mwrank_MordellWeil(SageObject):
 
         - ``ok`` (bool) -- ``True`` if and only if the saturation was
           provably successful at all primes attempted.  If the default
-          was used for ``max_prime`` and no warning was output about
-          the computed saturation bound being too high, then ``True``
-          indicates that the subgroup is saturated at *all*
-          primes.
+          was used for ``max_prime``, then ``True`` indicates that the
+          subgroup is saturated at *all* primes.
 
         - ``index`` (int) -- the index of the group generated by the
           original points in their saturation.
 
         - ``unsatlist`` (list of ints) -- list of primes at which
-          saturation could not be proved or achieved.  Increasing the
-          precision should correct this, since it happens when
-          a linear combination of the points appears to be a multiple
-          of `p` but cannot be divided by `p`.  (Note that ``eclib``
-          uses floating point methods based on elliptic logarithms to
-          divide points.)
+          saturation could not be proved or achieved.
 
         .. note::
 
-           We emphasize that if this function returns ``True`` as the
-           first return argument (``ok``), and if the default was used for the
-           parameter ``max_prime``, then the points in the basis after
-           calling this function are saturated at *all* primes,
-           i.e., saturating at the primes up to ``max_prime`` are
-           sufficient to saturate at all primes.  Note that the
-           function might not have needed to saturate at all primes up
-           to ``max_prime``.  It has worked out what prime you need to
-           saturate up to, and that prime might be smaller than ``max_prime``.
+          In versions up to v20190909, ``eclib`` used floating point
+          methods based on elliptic logarithms to divide points, and
+          did not compute the precision necessary, which could casue
+          failures. Since v20210310, ``eclib`` uses exact method based
+          on division polynomials, which should mean that such
+          failures does not happen.
 
         .. note::
 
-           Currently (May 2010), this does not remember the result of
-           calling :meth:`search()`.  So calling :meth:`search()` up
-           to height 20 then calling :meth:`saturate()` results in
-           another search up to height 18.
+           We emphasize that if this function returns ``True`` as the
+           first return argument (``ok``), and if the default was used
+           for the parameter ``max_prime``, then the points in the
+           basis after calling this function are saturated at *all*
+           primes, i.e., saturating at the primes up to ``max_prime``
+           are sufficient to saturate at all primes.  Note that the
+           function computes an upper bound for the index of
+           saturation, and does no work for primes greater than this
+           even if ``max_prime`` is larger.
 
         EXAMPLES::
 
@@ -1160,7 +1129,7 @@ class mwrank_MordellWeil(SageObject):
         automatic saturation at this stage we set the parameter
         ``sat`` to 0 (which is in fact the default)::
 
-            sage: EQ.process([[1547, -2967, 343], [2707496766203306, 864581029138191, 2969715140223272], [-13422227300, -49322830557, 12167000000]], sat=0)
+            sage: EQ.process([[1547, -2967, 343], [2707496766203306, 864581029138191, 2969715140223272], [-13422227300, -49322830557, 12167000000]], saturation_bound=0)
             P1 = [1547:-2967:343]         is generator number 1
             P2 = [2707496766203306:864581029138191:2969715140223272]      is generator number 2
             P3 = [-13422227300:-49322830557:12167000000]          is generator number 3
@@ -1172,12 +1141,12 @@ class mwrank_MordellWeil(SageObject):
         Now we saturate at `p=2`, and gain index 2::
 
             sage: EQ.saturate(2)  # points were not 2-saturated
-            saturating basis...Saturation index bound = 93
-            WARNING: saturation at primes p > 2 will not be done;
+            saturating basis...Saturation index bound (for points of good reduction) = 93
+            Only p-saturating for p up to given value 2.
             ...
             Gained index 2
             New regulator =  93.857...
-            (False, 2, '[ ]')
+            (True, 2, '[ ]')
             sage: EQ
             Subgroup of Mordell-Weil group: [[-2:3:1], [2707496766203306:864581029138191:2969715140223272], [-13422227300:-49322830557:12167000000]]
             sage: EQ.regulator()
@@ -1186,12 +1155,12 @@ class mwrank_MordellWeil(SageObject):
         Now we saturate at `p=3`, and gain index 3::
 
             sage: EQ.saturate(3)  # points were not 3-saturated
-            saturating basis...Saturation index bound = 46
-            WARNING: saturation at primes p > 3 will not be done;
+            saturating basis...Saturation index bound (for points of good reduction) = 46
+            Only p-saturating for p up to given value 3.
             ...
             Gained index 3
             New regulator =  10.428...
-            (False, 3, '[ ]')
+            (True, 3, '[ ]')
             sage: EQ
             Subgroup of Mordell-Weil group: [[-2:3:1], [-14:25:8], [-13422227300:-49322830557:12167000000]]
             sage: EQ.regulator()
@@ -1200,12 +1169,12 @@ class mwrank_MordellWeil(SageObject):
         Now we saturate at `p=5`, and gain index 5::
 
             sage: EQ.saturate(5)  # points were not 5-saturated
-            saturating basis...Saturation index bound = 15
-            WARNING: saturation at primes p > 5 will not be done;
+            saturating basis...Saturation index bound (for points of good reduction) = 15
+            Only p-saturating for p up to given value 5.
             ...
             Gained index 5
             New regulator =  0.417...
-            (False, 5, '[ ]')
+            (True, 5, '[ ]')
             sage: EQ
             Subgroup of Mordell-Weil group: [[-2:3:1], [-14:25:8], [1:-1:1]]
             sage: EQ.regulator()
@@ -1215,7 +1184,8 @@ class mwrank_MordellWeil(SageObject):
         the points are now provably saturated at all primes::
 
             sage: EQ.saturate()   # points are now saturated
-            saturating basis...Saturation index bound = 3
+            saturating basis...Saturation index bound (for points of good reduction) = 3
+            Tamagawa index primes are [ ]
             Checking saturation at [ 2 3 ]
             Checking 2-saturation
             Points were proved 2-saturated (max q used = 11)
@@ -1229,7 +1199,7 @@ class mwrank_MordellWeil(SageObject):
 
             sage: E = mwrank_EllipticCurve([0,0,1,-7,6])
             sage: EQ = mwrank_MordellWeil(E)
-            sage: EQ.process([[1547, -2967, 343], [2707496766203306, 864581029138191, 2969715140223272], [-13422227300, -49322830557, 12167000000]], sat=5)
+            sage: EQ.process([[1547, -2967, 343], [2707496766203306, 864581029138191, 2969715140223272], [-13422227300, -49322830557, 12167000000]], saturation_bound=5)
             P1 = [1547:-2967:343]         is generator number 1
             ...
             Gained index 5, new generators = [ [-2:3:1] [-14:25:8] [1:-1:1] ]
@@ -1242,7 +1212,8 @@ class mwrank_MordellWeil(SageObject):
         verify that full saturation has been done::
 
             sage: EQ.saturate()
-            saturating basis...Saturation index bound = 3
+            saturating basis...Saturation index bound (for points of good reduction) = 3
+            Tamagawa index primes are [ ]
             Checking saturation at [ 2 3 ]
             Checking 2-saturation
             Points were proved 2-saturated (max q used = 11)
@@ -1255,8 +1226,9 @@ class mwrank_MordellWeil(SageObject):
         index of the points in their saturation is at most 3, then
         proves saturation at 2 and at 3, by reducing the points modulo
         all primes of good reduction up to 11, respectively 13.
+
         """
-        ok, index, unsat = self.__mw.saturate(int(max_prime), odd_primes_only)
+        ok, index, unsat = self.__mw.saturate(int(max_prime), int(min_prime))
         return bool(ok), int(str(index)), unsat
 
     def search(self, height_limit=18, verbose=False):
@@ -1271,9 +1243,9 @@ class mwrank_MordellWeil(SageObject):
 
         .. note::
 
-          On 32-bit machines, this *must* be < 21.48 else
+          On 32-bit machines, this *must* be < 21.48 (`31\log(2)`) else
           `\exp(h_{\text{lim}}) > 2^{31}` and overflows.  On 64-bit machines, it
-          must be *at most* 43.668.  However, this bound is a logarithmic
+          must be *at most* 43.668  (`63\log(2)`) .  However, this bound is a logarithmic
           bound and increasing it by just 1 increases the running time
           by (roughly) `\exp(1.5)=4.5`, so searching up to even 20
           takes a very long time.
@@ -1320,8 +1292,10 @@ class mwrank_MordellWeil(SageObject):
             Subgroup of Mordell-Weil group: [[4413270:10381877:27000]]
         """
         height_limit = float(height_limit)
-        if height_limit >= 21.4:        # TODO: docstring says 21.48 (for 32-bit machines; what about 64-bit...?)
-            raise ValueError("The height limit must be < 21.4.")
+        int_bits = sys.maxsize.bit_length()
+        max_height_limit = int_bits * 0.693147 # log(2.0) = 0.693147 approx
+        if height_limit >= max_height_limit:
+            raise ValueError("The height limit must be < {} = {}log(2) on a {}-bit machine.".format(max_height_limit, int_bits, int_bits+1))
 
         moduli_option = 0  # Use Stoll's sieving program... see strategies in ratpoints-1.4.c
 
@@ -1352,5 +1326,4 @@ class mwrank_MordellWeil(SageObject):
             [[1, -1, 1], [-2, 3, 1], [-14, 25, 8]]
 
         """
-        L = eval(self.__mw.getbasis().replace(":",","))
-        return [[Integer(x), Integer(y), Integer(z)] for (x,y,z) in L]
+        return self.__mw.getbasis()
diff --git a/src/sage/libs/eclib/mwrank.pyx b/src/sage/libs/eclib/mwrank.pyx
index b82831d..ce5090c 100644
--- a/src/sage/libs/eclib/mwrank.pyx
+++ b/src/sage/libs/eclib/mwrank.pyx
@@ -28,6 +28,7 @@ from cysignals.signals cimport sig_on, sig_off
 from sage.cpython.string cimport char_to_str, str_to_bytes
 from sage.cpython.string import FS_ENCODING
 from sage.libs.eclib cimport bigint, Curvedata, mw, two_descent
+from sage.rings.all import Integer
 
 cdef extern from "wrap.cpp":
     ### misc functions ###
@@ -55,8 +56,8 @@ cdef extern from "wrap.cpp":
     char* mw_getbasis(mw* m)
     double mw_regulator(mw* m)
     int mw_rank(mw* m)
-    int mw_saturate(mw* m, bigint* index, char** unsat,
-                    long sat_bd, int odd_primes_only)
+    int mw_saturate(mw* m, long* index, char** unsat,
+                    long sat_bd, long sat_low_bd)
     void mw_search(mw* m, char* h_lim, int moduli_option, int verb)
 
     ### two_descent ###
@@ -67,8 +68,7 @@ cdef extern from "wrap.cpp":
     long two_descent_get_rank(two_descent* t)
     long two_descent_get_rank_bound(two_descent* t)
     long two_descent_get_selmer_rank(two_descent* t)
-    void two_descent_saturate(two_descent* t, long sat_bd)
-
+    void two_descent_saturate(two_descent* t, long sat_bd, long sat_low_bd)
 
 cdef object string_sigoff(char* s):
     sig_off()
@@ -445,7 +445,6 @@ cdef class _Curvedata:   # cython class wrapping eclib's Curvedata class
             -1269581104000000
         """
         sig_on()
-        from sage.rings.all import Integer
         return Integer(string_sigoff(Curvedata_getdiscr(self.x)))
 
     def conductor(self):
@@ -467,7 +466,6 @@ cdef class _Curvedata:   # cython class wrapping eclib's Curvedata class
             126958110400
         """
         sig_on()
-        from sage.rings.all import Integer
         return Integer(string_sigoff(Curvedata_conductor(self.x)))
 
     def isogeny_class(self, verbose=False):
@@ -503,6 +501,36 @@ cdef class _Curvedata:   # cython class wrapping eclib's Curvedata class
 
 ############# _mw #################
 
+def parse_point_list(s):
+    r"""
+    Parse a string representing a list of points.
+
+    INPUT:
+
+    - ``s`` (string) -- string representation of a list of points, for
+      example '[]', '[[1:2:3]]', or '[[1:2:3],[4:5:6]]'.
+
+    OUTPUT:
+
+    (list)  a list of triples of integers, for example [], [[1,2,3]], [[1,2,3],[4,5,6]].
+
+    EXAMPLES::
+
+        sage: from sage.libs.eclib.mwrank import parse_point_list
+        sage: parse_point_list('[]')
+        []
+        sage: parse_point_list('[[1:2:3]]')
+        [[1, 2, 3]]
+        sage: parse_point_list('[[1:2:3],[4:5:6]]')
+        [[1, 2, 3], [4, 5, 6]]
+
+    """
+    s = s.replace(":", ",").replace(" ", "")
+    if s == '[]':
+        return []
+    pts = s[2:-2].split('],[')
+    return [[Integer(x) for x in pt.split(",")] for pt in pts]
+
 cdef class _mw:
     """
     Cython class wrapping eclib's mw class.
@@ -561,72 +589,37 @@ cdef class _mw:
             sage: EQ.search(1)
             P1 = [0:1:0]         is torsion point, order 1
             P1 = [-3:0:1]         is generator number 1
-            ...
-            P4 = [12:35:27]      = 1*P1 + -1*P2 + -1*P3 (mod torsion)
-
-        The previous command produces the following output::
-
-            P1 = [0:1:0]         is torsion point, order 1
-            P1 = [-3:0:1]         is generator number 1
-            saturating up to 20...Checking 2-saturation
-            Points have successfully been 2-saturated (max q used = 7)
-            Checking 3-saturation
-            Points have successfully been 3-saturated (max q used = 7)
-            Checking 5-saturation
-            Points have successfully been 5-saturated (max q used = 23)
-            Checking 7-saturation
-            Points have successfully been 7-saturated (max q used = 41)
-            Checking 11-saturation
-            Points have successfully been 11-saturated (max q used = 17)
-            Checking 13-saturation
-            Points have successfully been 13-saturated (max q used = 43)
-            Checking 17-saturation
-            Points have successfully been 17-saturated (max q used = 31)
-            Checking 19-saturation
-            Points have successfully been 19-saturated (max q used = 37)
+            saturating up to 20...Saturation index bound (for points of good reduction)  = 3
+            Reducing saturation bound from given value 20 to computed index bound 3
+            Checking saturation at [ 2 3 ]
+            Checking 2-saturation 
+            Points were proved 2-saturated (max q used = 7)
+            Checking 3-saturation 
+            Points were proved 3-saturated (max q used = 7)
             done
             P2 = [-2:3:1]         is generator number 2
-            saturating up to 20...Checking 2-saturation
+            saturating up to 20...Saturation index bound (for points of good reduction)  = 4
+            Reducing saturation bound from given value 20 to computed index bound 4
+            Checking saturation at [ 2 3 ]
+            Checking 2-saturation 
             possible kernel vector = [1,1]
             This point may be in 2E(Q): [14:-52:1]
-            ...and it is!
+            ...and it is! 
             Replacing old generator #1 with new generator [1:-1:1]
+            Reducing index bound from 4 to 2
             Points have successfully been 2-saturated (max q used = 7)
             Index gain = 2^1
-            Checking 3-saturation
-            Points have successfully been 3-saturated (max q used = 13)
-            Checking 5-saturation
-            Points have successfully been 5-saturated (max q used = 67)
-            Checking 7-saturation
-            Points have successfully been 7-saturated (max q used = 53)
-            Checking 11-saturation
-            Points have successfully been 11-saturated (max q used = 73)
-            Checking 13-saturation
-            Points have successfully been 13-saturated (max q used = 103)
-            Checking 17-saturation
-            Points have successfully been 17-saturated (max q used = 113)
-            Checking 19-saturation
-            Points have successfully been 19-saturated (max q used = 47)
-            done (index = 2).
+            done, index = 2.
             Gained index 2, new generators = [ [1:-1:1] [-2:3:1] ]
             P3 = [-14:25:8]       is generator number 3
-            saturating up to 20...Checking 2-saturation
-            Points have successfully been 2-saturated (max q used = 11)
-            Checking 3-saturation
-            Points have successfully been 3-saturated (max q used = 13)
-            Checking 5-saturation
-            Points have successfully been 5-saturated (max q used = 71)
-            Checking 7-saturation
-            Points have successfully been 7-saturated (max q used = 101)
-            Checking 11-saturation
-            Points have successfully been 11-saturated (max q used = 127)
-            Checking 13-saturation
-            Points have successfully been 13-saturated (max q used = 151)
-            Checking 17-saturation
-            Points have successfully been 17-saturated (max q used = 139)
-            Checking 19-saturation
-            Points have successfully been 19-saturated (max q used = 179)
-            done (index = 1).
+            saturating up to 20...Saturation index bound (for points of good reduction)  = 3
+            Reducing saturation bound from given value 20 to computed index bound 3
+            Checking saturation at [ 2 3 ]
+            Checking 2-saturation 
+            Points were proved 2-saturated (max q used = 11)
+            Checking 3-saturation 
+            Points were proved 3-saturated (max q used = 13)
+            done, index = 1.
             P4 = [-1:3:1]        = -1*P1 + -1*P2 + -1*P3 (mod torsion)
             P4 = [0:2:1]         = 2*P1 + 0*P2 + 1*P3 (mod torsion)
             P4 = [2:13:8]        = -3*P1 + 1*P2 + -1*P3 (mod torsion)
@@ -687,7 +680,7 @@ cdef class _mw:
         sig_on()
         return string_sigoff(mw_getbasis(self.x))
 
-    def process(self, point, sat=0):
+    def process(self, point, saturation_bound=0):
         """
         Processes the given point, adding it to the mw group.
 
@@ -697,10 +690,12 @@ cdef class _mw:
           An ``ArithmeticError`` is raised if the point is not on the
           curve.
 
-        - ``sat`` (int, default 0) --saturate at primes up to ``sat``.
-          No saturation is done if ``sat=0``.  (Note that it is more
-          efficient to add several points at once and then saturate
-          just once at the end).
+        - ``saturation_bound`` (int, default 0) --saturate at primes up to ``saturation_bound``.
+          No saturation is done if ``saturation_bound=0``.  If ``saturation_bound=-1`` then
+          saturation is done at all primes, by computing a bound on
+          the saturation index.  Note that it is more efficient to add
+          several points at once and then saturate just once at the
+          end.
 
         .. NOTE::
 
@@ -746,7 +741,7 @@ cdef class _mw:
         cdef _bigint x,y,z
         sig_on()
         x,y,z = _bigint(point[0]), _bigint(point[1]), _bigint(point[2])
-        r = mw_process(self.curve, self.x, x.x, y.x, z.x, sat)
+        r = mw_process(self.curve, self.x, x.x, y.x, z.x, saturation_bound)
         sig_off()
         if r != 0:
             raise ArithmeticError("point (=%s) not on curve." % point)
@@ -757,8 +752,8 @@ cdef class _mw:
 
         OUTPUT:
 
-        (string) String representation of the points in the basis of
-        the mw group.
+        (list) list of integer triples giving the projective
+        coordinates of the points in the basis.
 
         EXAMPLES::
 
@@ -768,13 +763,13 @@ cdef class _mw:
             sage: EQ = _mw(E)
             sage: EQ.search(3)
             sage: EQ.getbasis()
-            '[[0:-1:1], [-1:1:1]]'
+            [[0, -1, 1], [-1, 1, 1]]
             sage: EQ.rank()
             2
         """
         sig_on()
         s = string_sigoff(mw_getbasis(self.x))
-        return s
+        return parse_point_list(s)
 
     def regulator(self):
         """
@@ -797,7 +792,7 @@ cdef class _mw:
             sage: EQ = _mw(E)
             sage: EQ.search(3)
             sage: EQ.getbasis()
-            '[[0:-1:1], [-1:1:1]]'
+            [[0, -1, 1], [-1, 1, 1]]
             sage: EQ.rank()
             2
             sage: EQ.regulator()
@@ -824,39 +819,54 @@ cdef class _mw:
             sage: EQ = _mw(E)
             sage: EQ.search(3)
             sage: EQ.getbasis()
-            '[[0:-1:1], [-1:1:1]]'
+            [[0, -1, 1], [-1, 1, 1]]
             sage: EQ.rank()
             2
         """
         sig_on()
         r = mw_rank(self.x)
         sig_off()
-        from sage.rings.all import Integer
         return Integer(r)
 
-    def saturate(self, int sat_bd=-1, int odd_primes_only=0):
+    def saturate(self, int sat_bd=-1, int sat_low_bd=2):
         """
         Saturates the current subgroup of the mw group.
 
         INPUT:
 
-        - ``sat_bnd`` (int, default -1) -- bound on primes at which to
-          saturate.  If -1 (default), compute a bound for the primes
-          which may not be saturated, and use that.
+        - ``sat_bnd`` (int, default -1) -- upper bound on primes at
+          which to saturate.  If -1 (default), compute a bound for the
+          primes which may not be saturated, and use that.  Otherwise,
+          the bound used is the minumum of the value of ``sat_bnd``
+          and the computed bound.
 
-        - ``odd_primes_only`` (bool, default ``False``) -- only do
-          saturation at odd primes.  (If the points have been found
-          via 2-descent they should already be 2-saturated.)
+        - ``sat_low_bd`` (int, default 2) -- only do saturation at
+          prime not less than this.  For exampe, if the points have
+          been found via 2-descent they should already be 2-saturated,
+          and ``sat_low_bd=3`` is appropriate.
 
         OUTPUT:
 
         (tuple) (success flag, index, list) The success flag will be 1
         unless something failed (usually an indication that the points
-        were not saturated but the precision is not high enough to
-        divide out successfully).  The index is the index of the mw
-        group before saturation in the mw group after.  The list is a
-        string representation of the primes at which saturation was
-        not proved or achieved.
+        were not saturated but eclib was not able to divide out
+        successfully).  The index is the index of the mw group before
+        saturation in the mw group after.  The list is a string
+        representation of the primes at which saturation was not
+        proved or achieved.
+
+        .. NOTE::
+
+        ``eclib`` will compute a bound on the saturation index.  If
+        the computed saturation bound is very large and ``sat_bnd`` is
+        -1, ``eclib`` may output a warning, but will still attempt to
+        saturate up to the computed bound.  If a positive value of
+        ``sat_bnd`` is given which is greater than the computed bound,
+        `p`-saturation will only be carried out for primes up to the
+        compated bound.  Setting ``sat_low_bnd`` to a value greater
+        than 2 allows for saturation to be done incrementally, or for
+        exactly one prime `p` by setting both ``sat_bd`` and
+        ``sat_low_bd`` to `p`.
 
         EXAMPLES::
 
@@ -872,34 +882,23 @@ cdef class _mw:
             sage: EQ
             [[-1:1:1]]
 
-        If we set the saturation bound at 2, then saturation will fail::
+        If we set the saturation bound at 2, then saturation will not
+        enlarge the basis, but the success flag is still 1 (True)
+        since we did not ask to check 3-saturation::
 
             sage: EQ = _mw(E)
             sage: EQ.process([494, -5720, 6859]) # 3 times another point
             sage: EQ.saturate(sat_bd=2)
-            Saturation index bound = 10
-            WARNING: saturation at primes p > 2 will not be done;
-            points may be unsaturated at primes between 2 and index bound
-            Failed to saturate MW basis at primes [ ]
-            (0, 1, '[ ]')
+            (1, 1, '[ ]')
             sage: EQ
             [[494:-5720:6859]]
 
-        The following output is also seen in the preceding example::
-
-            Saturation index bound = 10
-            WARNING: saturation at primes p > 2 will not be done;
-            points may be unsaturated at primes between 2 and index bound
-            Failed to saturate MW basis at primes [ ]
-
-
         """
-        cdef _bigint index
+        cdef long index
         cdef char* s
         cdef int ok
         sig_on()
-        index = _bigint()
-        ok = mw_saturate(self.x, index.x, &s, sat_bd, odd_primes_only)
+        ok = mw_saturate(self.x, &index, &s, sat_bd, sat_low_bd)
         unsat = string_sigoff(s)
         return ok, index, unsat
 
@@ -1094,7 +1093,6 @@ cdef class _two_descent:
         sig_on()
         r = two_descent_get_rank(self.x)
         sig_off()
-        from sage.rings.all import Integer
         return Integer(r)
 
     def getrankbound(self):
@@ -1128,7 +1126,6 @@ cdef class _two_descent:
         sig_on()
         r = two_descent_get_rank_bound(self.x)
         sig_off()
-        from sage.rings.all import Integer
         return Integer(r)
 
     def getselmer(self):
@@ -1161,7 +1158,6 @@ cdef class _two_descent:
         sig_on()
         r = two_descent_get_selmer_rank(self.x)
         sig_off()
-        from sage.rings.all import Integer
         return Integer(r)
 
     def ok(self):
@@ -1222,10 +1218,21 @@ cdef class _two_descent:
         """
         return two_descent_get_certain(self.x)
 
-    def saturate(self, saturation_bound=0):
+    def saturate(self, saturation_bound=0, lower=3):
         """
         Carries out saturation of the points found by a 2-descent.
 
+        INPUT:
+
+        - ``saturation_bound`` (int) -- an upper bound on the primes
+          `p` at which `p`-saturation will be carried out, or -1, in
+          which case ``eclib`` will compute an upper bound on the
+          saturation index.
+
+        - ``lower`` (int, default 3) -- do no `p`-saturation for `p`
+          less than this.  The default is 3 since the points found
+          during 2-descent will be 2-saturated.
+
         OUTPUT:
 
         None.
@@ -1257,7 +1264,7 @@ cdef class _two_descent:
             '[[1:-1:1], [-2:3:1], [-14:25:8]]'
         """
         sig_on()
-        two_descent_saturate(self.x, saturation_bound)
+        two_descent_saturate(self.x, saturation_bound, 3)
         sig_off()
 
     def getbasis(self):
diff --git a/src/sage/libs/eclib/newforms.pyx b/src/sage/libs/eclib/newforms.pyx
index b50b606..96263cd 100644
--- a/src/sage/libs/eclib/newforms.pyx
+++ b/src/sage/libs/eclib/newforms.pyx
@@ -140,6 +140,7 @@ cdef class ECModularSymbol:
 
         - ``nap`` - (int, default 1000): the number of ap of E to use
          in determining the normalisation of the modular symbols.
+         Note that eclib will increase this to 100*sqrt(N) if necessary.
 
         EXAMPLES::
 
diff --git a/src/sage/libs/eclib/t b/src/sage/libs/eclib/t
new file mode 100644
index 00000000..e69de29
--- /dev/null
+++ b/src/sage/libs/eclib/t
diff --git a/src/sage/libs/eclib/wrap.cpp b/src/sage/libs/eclib/wrap.cpp
index 58c18ab..28e6da8 100644
--- a/src/sage/libs/eclib/wrap.cpp
+++ b/src/sage/libs/eclib/wrap.cpp
@@ -178,11 +178,11 @@ int mw_rank(struct mw* m)
 }
 
 /* Returns index and unsat long array, which user must deallocate */
-int mw_saturate(struct mw* m, bigint* index, char** unsat,
-                       long sat_bd, int odd_primes_only)
+int mw_saturate(struct mw* m, long* index, char** unsat,
+                       long sat_bd, long sat_low_bd)
 {
   vector<long> v;
-  int s = m->saturate(*index, v, sat_bd, odd_primes_only);
+  int s = m->saturate(*index, v, sat_bd, sat_low_bd);
   ostringstream instore;
   instore << v;
   *unsat  = stringstream_to_char(instore);
@@ -236,9 +236,9 @@ long two_descent_get_certain(const two_descent* t)
   return t->getcertain();
 }
 
-void two_descent_saturate(struct two_descent* t, long sat_bd)
+void two_descent_saturate(struct two_descent* t, long sat_bd, long sat_low_bd)
 {
-  t->saturate(sat_bd);
+  t->saturate(sat_bd, sat_low_bd);
 }
 
 double two_descent_regulator(struct two_descent* t)
diff --git a/src/sage/schemes/elliptic_curves/ell_modular_symbols.py b/src/sage/schemes/elliptic_curves/ell_modular_symbols.py
index a32f64e..30a61e1 100644
--- a/src/sage/schemes/elliptic_curves/ell_modular_symbols.py
+++ b/src/sage/schemes/elliptic_curves/ell_modular_symbols.py
@@ -298,19 +298,27 @@ class ModularSymbolECLIB(ModularSymbol):
             sage: m(0)
             1/5
 
-        If ``nap`` is too small, the normalization in eclib may be incorrect.  See :trac:`31317`::
+        If ``nap`` is too small, the normalization in eclib used to be
+        incorrect (see :trac:`31317`), but since ``eclib`` version
+        v20210310 the value of ``nap`` is increased automatically by
+        ``eclib``::
 
             sage: from sage.schemes.elliptic_curves.ell_modular_symbols import ModularSymbolECLIB
             sage: E = EllipticCurve('1590g1')
             sage: m = ModularSymbolECLIB(E, sign=+1, nap=300)
             sage: [m(a/5) for a in [1..4]]
-            [1001/153, -1001/153, -1001/153, 1001/153]
+            [13/2, -13/2, -13/2, 13/2]
 
-        Those values are incorrect.  The correct values are::
+        These values are correct, and increasing ``nap`` has no
+        effect.  The correct values may verified by the numerical
+        implementation::
 
             sage: m = ModularSymbolECLIB(E, sign=+1, nap=400)
             sage: [m(a/5) for a in [1..4]]
             [13/2, -13/2, -13/2, 13/2]
+            sage: m = E.modular_symbol(implementation='num')
+            sage: [m(a/5) for a in [1..4]]
+            [13/2, -13/2, -13/2, 13/2]
 
         """
         from sage.libs.eclib.newforms import ECModularSymbol
diff --git a/src/sage/schemes/elliptic_curves/ell_rational_field.py b/src/sage/schemes/elliptic_curves/ell_rational_field.py
index a792afc..5a56389 100644
--- a/src/sage/schemes/elliptic_curves/ell_rational_field.py
+++ b/src/sage/schemes/elliptic_curves/ell_rational_field.py
@@ -779,7 +779,7 @@ class EllipticCurve_rational_field(EllipticCurve_number_field):
             sage: E = EllipticCurve('11a1')
             sage: EE = E.mwrank_curve()
             sage: EE
-            y^2+ y = x^3 - x^2 - 10*x - 20
+            y^2 + y = x^3 - x^2 - 10 x - 20
             sage: type(EE)
             <class 'sage.libs.eclib.interface.mwrank_EllipticCurve'>
             sage: EE.isogeny_class()
@@ -1283,22 +1283,21 @@ class EllipticCurve_rational_field(EllipticCurve_number_field):
             sage: [Mminus(1/i) for i in [1..11]]
             [0, 0, 1/2, 1/2, 0, 0, -1/2, -1/2, 0, 0, 0]
 
-        With the default 'eclib' implementation, if ``nap`` is too
-        small, the normalization may be computed incorrectly.  See
-        :trac:`31317`::
+        With older version of eclib, in the default 'eclib'
+        implementation, if ``nap`` is too small, the normalization may
+        be computed incorrectly (see :trac:`31317`).  This was fixed
+        in eclib version v20210310, since now eclib increase ``nap``
+        automatically. The following used to give incorrect results.
+        See :trac:`31443`::
 
             sage: E = EllipticCurve('1590g1')
             sage: m = E.modular_symbol(nap=300)
             sage: [m(a/5) for a in [1..4]]
-            [1001/153, -1001/153, -1001/153, 1001/153]
+            [13/2, -13/2, -13/2, 13/2]
 
-        Those values are incorrect.  The correct values may be
-        obtained by increasing ``nap``, as verified by the numerical
+        These values are correct, as verified by the numerical
         implementation::
 
-            sage: m = E.modular_symbol(nap=400)
-            sage: [m(a/5) for a in [1..4]]
-            [13/2, -13/2, -13/2, 13/2]
             sage: m = E.modular_symbol(implementation='num')
             sage: [m(a/5) for a in [1..4]]
             [13/2, -13/2, -13/2, 13/2]
@@ -2525,7 +2524,7 @@ class EllipticCurve_rational_field(EllipticCurve_number_field):
         assert reg.parent() is R
         return reg
 
-    def saturation(self, points, verbose=False, max_prime=0, odd_primes_only=False):
+    def saturation(self, points, verbose=False, max_prime=-1, min_prime=2):
         """
         Given a list of rational points on E, compute the saturation in
         E(Q) of the subgroup they generate.
@@ -2538,17 +2537,24 @@ class EllipticCurve_rational_field(EllipticCurve_number_field):
         -  ``verbose (bool)`` - (default: ``False``), if ``True``, give
            verbose output
 
-        -  ``max_prime (int)`` - (default: 0), saturation is
-           performed for all primes up to max_prime. If max_prime==0,
-           perform saturation at *all* primes, i.e., compute the true
-           saturation.
+        - ``max_prime`` (int, default -1) -- If `-1` (the default), an
+          upper bound is computed for the primes at which the subgroup
+          may not be saturated, and saturation is performed for all
+          primes up to this bound.  Otherwise, the bound used is the
+          minimum of ``max_prime`` and the computed bound.
 
-        -  ``odd_primes_only (bool)`` - only do saturation at
-           odd primes
+        - ``min_prime (int)`` - (default: 2), only do `p`-saturation
+            at primes `p` greater than or equal to this.
 
+        .. note::
 
-        OUTPUT:
+           To saturate at a single prime `p`, set ``max_prime`` and
+           ``min_prime`` both to `p`.  One situation where this is
+           useful is after mapping saturated points from another
+           elliptic curve by a `p`-isogeny, since the images may not
+           be `p`-saturated but with be saturated at all other primes.
 
+        OUTPUT:
 
         -  ``saturation (list)`` - points that form a basis for
            the saturation
@@ -2559,12 +2565,32 @@ class EllipticCurve_rational_field(EllipticCurve_number_field):
         -  ``regulator (real with default precision)`` -
            regulator of saturated points.
 
+        ALGORITHM: Uses Cremona's ``eclib`` package, which computes a
+        bound on the saturation index.  To `p`-saturate, or prove
+        `p`-saturation, we consider the reductions of the points
+        modulo primes `q` of good reduction such that `E(\FF_q)` has
+        order divisible by `p`.
+
+        .. note::
+
+           In versons of ``eclib`` up to ``v20190909``, division of
+           points in ``eclib`` was done using floating point methods,
+           without automatic handling of precision, so that
+           `p`-saturation sometimes failed unless
+           ``mwrank_set_precision()`` was called in advance with a
+           suitably high bit precision.  Since version ``v20210310``
+           of ``eclib``, division is done using exact methods based on
+           division polynomials, and `p`-saturation cannot fail in
+           this way.
+
+        .. note::
+
+           The computed index of saturation may be large, in which
+           case saturation may take a long time.  For example, the
+           rank 4 curve ``EllipticCurve([0,1,1,-9872,374262])`` has a
+           saturation index bound of 86682 and takes around 15 minutes
+           to prove saturation.
 
-        ALGORITHM: Uses Cremona's ``mwrank`` package. With ``max_prime=0``,
-        we call ``mwrank`` with successively larger prime bounds until the full
-        saturation is provably found. The results of saturation at the
-        previous primes is stored in each case, so this should be
-        reasonably fast.
 
         EXAMPLES::
 
@@ -2577,7 +2603,9 @@ class EllipticCurve_rational_field(EllipticCurve_number_field):
 
         TESTS:
 
-        See :trac:`10590`.  This example would loop forever at default precision::
+        See :trac:`10590`.  With ``eclib`` versions up to
+        ``v20190909``, this example would loop forever at default
+        precision.  Since version ``v20210310`` it runs fine::
 
             sage: E = EllipticCurve([1, 0, 1, -977842, -372252745])
             sage: P = E([-192128125858676194585718821667542660822323528626273/336995568430319276695106602174283479617040716649, 70208213492933395764907328787228427430477177498927549075405076353624188436/195630373799784831667835900062564586429333568841391304129067339731164107, 1])
@@ -2585,7 +2613,7 @@ class EllipticCurve_rational_field(EllipticCurve_number_field):
             113.302910926080
             sage: E.saturation([P])
             ([(-192128125858676194585718821667542660822323528626273/336995568430319276695106602174283479617040716649 : 70208213492933395764907328787228427430477177498927549075405076353624188436/195630373799784831667835900062564586429333568841391304129067339731164107 : 1)], 1, 113.302910926080)
-            sage: (Q,), ind, reg = E.saturation([2*P])  # needs higher precision, handled by eclib
+            sage: (Q,), ind, reg = E.saturation([2*P])
             sage: 2*Q == 2*P
             True
             sage: ind
@@ -2634,36 +2662,16 @@ class EllipticCurve_rational_field(EllipticCurve_number_field):
         c = Emin.mwrank_curve()
         from sage.libs.eclib.all import mwrank_MordellWeil
         mw = mwrank_MordellWeil(c, verbose)
-        mw.process(v)
-        repeat_until_saturated = False
-        if max_prime == 0:
-            repeat_until_saturated = True
-            max_prime = 9973
-        from sage.libs.all import mwrank_get_precision, mwrank_set_precision
-        prec0 = mwrank_get_precision()
-        prec = 100
-        if prec0<prec:
-            mwrank_set_precision(prec)
-        else:
-            prec = prec0
-        while True:
-            ok, index, unsat = mw.saturate(max_prime=max_prime, odd_primes_only = odd_primes_only)
-            reg = mw.regulator()
-            if ok or not repeat_until_saturated:
-                break
-            max_prime = arith.next_prime(max_prime + 1000)
-            prec += 50
-            mwrank_set_precision(prec)
-        if prec != prec0:
-            mwrank_set_precision(prec0)
-        sat = mw.points()
-        sat = [Emin(P) for P in sat]
+        mw.process(v) # by default, this does no saturation yet
+        ok, index, unsat = mw.saturate(max_prime=max_prime, min_prime = min_prime)
+        if not ok:
+            print("Failed to saturate failed at the primes {}".format(unsat))
+        sat = [Emin(P) for P in mw.points()]
         if not minimal:
             phi_inv = ~phi
             sat = [phi_inv(P) for P in sat]
         reg = self.regulator_of_points(sat)
-        return sat, index, R(reg)
-
+        return sat, index, reg
 
     def CPS_height_bound(self):
         r"""