Drop merged patch

3 files changed, 4 insertions, 1416 deletions

diff --git a/.SRCINFO b/.SRCINFO
index 0e0cb0c102e5..a1ad2406ab52 100644
--- a/.SRCINFO
+++ b/.SRCINFO
@@ -1,6 +1,6 @@
 pkgbase = sagemath-git
 	pkgdesc = Open Source Mathematics Software, free alternative to Magma, Maple, Mathematica, and Matlab
-	pkgver = 9.4.beta3.r0.ga60179ab6b
+	pkgver = 9.4.beta4.r0.g473cd41f19
 	pkgrel = 1
 	url = http://www.sagemath.org
 	arch = x86_64
@@ -101,12 +101,10 @@ pkgbase = sagemath-git
 	source = latte-count.patch
 	source = test-optional.patch
 	source = sagemath-lrcalc2.patch
-	source = sagemath-eclib-20210310.patch
 	sha256sums = SKIP
 	sha256sums = c100a61c8dfade43bebc622a363abcb3d935a2f40958371ad87a9eb00689f8b0
 	sha256sums = 88e944f23c3b2391dc2e9f9be8e1131152d837dc8c829dfc714663869a272e81
 	sha256sums = af984186f852d2847d770a18fb6822296c50a652dbf55a1ed59d27517c3d3ee4
 	sha256sums = 240ac4c29d96d56407a20e1b7f9846e342a7eb2bb4edd6e5c86b3b5a8ff462f9
-	sha256sums = e7b31f5e7ea88681c6eda41e5a74a2859a12dd128e75c00db3cfbd1d8ddf080d
 
 pkgname = sagemath-git
diff --git a/PKGBUILD b/PKGBUILD
index defa4e5e682c..54bf65e5375b 100644
--- a/PKGBUILD
+++ b/PKGBUILD
@@ -6,7 +6,7 @@
 # Contributor: Stefan Husmann <stefan-husmann at t-online dot de>
 
 pkgname=sagemath-git
-pkgver=9.4.beta3.r0.ga60179ab6b
+pkgver=9.4.beta4.r0.g473cd41f19
 pkgrel=1
 pkgdesc='Open Source Mathematics Software, free alternative to Magma, Maple, Mathematica, and Matlab'
 arch=(x86_64)
@@ -41,14 +41,12 @@ source=(git://git.sagemath.org/sage.git#branch=develop
         sagemath-optional-packages.patch
         latte-count.patch
         test-optional.patch
-        sagemath-lrcalc2.patch
-        sagemath-eclib-20210310.patch)
+        sagemath-lrcalc2.patch)
 sha256sums=('SKIP'
             'c100a61c8dfade43bebc622a363abcb3d935a2f40958371ad87a9eb00689f8b0'
             '88e944f23c3b2391dc2e9f9be8e1131152d837dc8c829dfc714663869a272e81'
             'af984186f852d2847d770a18fb6822296c50a652dbf55a1ed59d27517c3d3ee4'
-            '240ac4c29d96d56407a20e1b7f9846e342a7eb2bb4edd6e5c86b3b5a8ff462f9'
-            'e7b31f5e7ea88681c6eda41e5a74a2859a12dd128e75c00db3cfbd1d8ddf080d')
+            '240ac4c29d96d56407a20e1b7f9846e342a7eb2bb4edd6e5c86b3b5a8ff462f9')
 
 pkgver() {
   cd sage
@@ -61,8 +59,6 @@ prepare(){
 # Upstream patches
 # Replace lrcalc.pyx with a wrapper over lrcalc's python bindings https://trac.sagemath.org/ticket/31355
   patch -p1 -i ../sagemath-lrcalc2.patch
-# Fix build with eclib 20210310 https://trac.sagemath.org/ticket/31443
-  patch -p1 -i ../sagemath-eclib-20210310.patch
 
 # Arch-specific patches
 # assume all optional packages are installed
diff --git a/sagemath-eclib-20210310.patch b/sagemath-eclib-20210310.patch
deleted file mode 100644
index bea233239119..000000000000
--- a/sagemath-eclib-20210310.patch
+++ /dev/null
@@ -1,1406 +0,0 @@
-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"""