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ports/math/sage/files/patch-src_sage_libs_eclib_interface.py
Thierry Thomas 886efe447a Fix build, with patch from John Cremona. 
Reported by:	pkg-fallout
2021-03-17 16:58:21 +00:00

699 lines
32 KiB
Python
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--- src/sage/libs/eclib/interface.py.orig 2021-03-16 21:40:43 UTC
+++ src/sage/libs/eclib/interface.py
@@ -21,18 +21,17 @@ 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"""
The :class:`mwrank_EllipticCurve` class represents an elliptic
@@ -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)
@@ -902,21 +842,20 @@ class mwrank_MordellWeil(SageObject):
"""
return "Subgroup of Mordell-Weil group: %s"%self.__mw
- def process(self, v, sat=0):
- """
+ def process(self, v, saturation_bound=0):
+ """Process points in the list ``v``.
+
This function allows one to add points to a :class:`mwrank_MordellWeil` object.
- Process points in the list ``v``, with saturation at primes up to
- ``sat``. If ``sat`` is zero (the default), do no saturation.
-
INPUT:
- ``v`` (list of 3-tuples or lists of ints or Integers) -- a
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()
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