Sage interface to Cremona’s eclib
library (also known as mwrank
)¶
This is the Sage interface to John Cremona’s eclib
C++ library for
arithmetic on elliptic curves. The classes defined in this module
give Sage interpreterlevel access to some of the functionality of
eclib
. For most purposes, it is not necessary to directly use these
classes. Instead, one can create an
EllipticCurve
and call methods that are implemented using this module.
Note
This interface is a direct librarylevel interface to eclib
,
including the 2descent program mwrank
.

class
sage.libs.eclib.interface.
mwrank_EllipticCurve
(ainvs, verbose=False)¶ Bases:
sage.structure.sage_object.SageObject
The
mwrank_EllipticCurve
class represents an elliptic curve using theCurvedata
class fromeclib
, called here an ‘mwrank elliptic curve’.Create the mwrank elliptic curve with invariants
ainvs
, which is a list of 5 or less integers \(a_1\), \(a_2\), \(a_3\), \(a_4\), and \(a_5\).If strictly less than 5 invariants are given, then the first ones are set to 0, so, e.g.,
[3,4]
means \(a_1=a_2=a_3=0\) and \(a_4=3\), \(a_5=4\).INPUT:
ainvs
(list or tuple) – a list of 5 or less integers, the coefficients of a nonsingular Weierstrass equation.verbose
(bool, defaultFalse
) – verbosity flag. IfTrue
, then all Selmer group computations will be verbose.
EXAMPLES:
We create the elliptic curve \(y^2 + y = x^3 + x^2  2x\):
sage: e = mwrank_EllipticCurve([0, 1, 1, 2, 0]) sage: e.ainvs() [0, 1, 1, 2, 0]
This example illustrates that omitted \(a\)invariants default to \(0\):
sage: e = mwrank_EllipticCurve([3, 4]) sage: e y^2 = x^3 + 3*x  4 sage: e.ainvs() [0, 0, 0, 3, 4]
The entries of the input list are coerced to
int
. If this is impossible, then an error is raised:sage: e = mwrank_EllipticCurve([3, 4.8]); e Traceback (most recent call last): ... TypeError: ainvs must be a list or tuple of integers.
When you enter a singular model you get an exception:
sage: e = mwrank_EllipticCurve([0, 0]) Traceback (most recent call last): ... ArithmeticError: Invariants (= 0,0,0,0,0) do not describe an elliptic curve.

CPS_height_bound
()¶ Return the CremonaPrickettSiksek height bound. This is a floating point number \(B\) such that if \(P\) is a point on the curve, then the naive logarithmic height \(h(P)\) is less than \(B+\hat{h}(P)\), where \(\hat{h}(P)\) is the canonical height of \(P\).
Warning
We assume the model is minimal!
EXAMPLES:
sage: E = mwrank_EllipticCurve([0, 0, 0, 1002231243161, 0]) sage: E.CPS_height_bound() 14.163198527061496 sage: E = mwrank_EllipticCurve([0,0,1,7,6]) sage: E.CPS_height_bound() 0.0

ainvs
()¶ Returns the \(a\)invariants of this mwrank elliptic curve.
EXAMPLES:
sage: E = mwrank_EllipticCurve([0,0,1,1,0]) sage: E.ainvs() [0, 0, 1, 1, 0]

certain
()¶ Returns
True
if the lasttwo_descent()
call provably correctly computed the rank. Iftwo_descent()
hasn’t been called, then it is first called bycertain()
using the default parameters.The result is
True
if and only if the results of the methodsrank()
andrank_bound()
are equal.EXAMPLES:
A 2descent does not determine \(E(\QQ)\) with certainty for the curve \(y^2 + y = x^3  x^2  120x  2183\):
sage: E = mwrank_EllipticCurve([0, 1, 1, 120, 2183]) sage: E.two_descent(False) ... sage: E.certain() False sage: E.rank() 0
The previous value is only a lower bound; the upper bound is greater:
sage: E.rank_bound() 2
In fact the rank of \(E\) is actually 0 (as one could see by computing the \(L\)function), but Sha has order 4 and the 2torsion is trivial, so mwrank cannot conclusively determine the rank in this case.

conductor
()¶ Return the conductor of this curve, computed using Cremona’s implementation of Tate’s algorithm.
Note
This is independent of PARI’s.
EXAMPLES:
sage: E = mwrank_EllipticCurve([1, 1, 0, 6958, 224588]) sage: E.conductor() 2310

gens
()¶ Return a list of the generators for the MordellWeil group.
EXAMPLES:
sage: E = mwrank_EllipticCurve([0, 0, 1, 1, 0]) sage: E.gens() [[0, 1, 1]]

isogeny_class
(verbose=False)¶ Returns the isogeny class of this mwrank elliptic curve.
EXAMPLES:
sage: E = mwrank_EllipticCurve([0,1,1,0,0]) sage: E.isogeny_class() ([[0, 1, 1, 0, 0], [0, 1, 1, 10, 20], [0, 1, 1, 7820, 263580]], [[0, 5, 0], [5, 0, 5], [0, 5, 0]])

rank
()¶ Returns the rank of this curve, computed using
two_descent()
.In general this may only be a lower bound for the rank; an upper bound may be obtained using the function
rank_bound()
. To test whether the value has been proved to be correct, use the methodcertain()
.EXAMPLES:
sage: E = mwrank_EllipticCurve([0, 1, 0, 900, 10098]) sage: E.rank() 0 sage: E.certain() True
sage: E = mwrank_EllipticCurve([0, 1, 1, 929, 10595]) sage: E.rank() 0 sage: E.certain() False

rank_bound
()¶ Returns an upper bound for the rank of this curve, computed using
two_descent()
.If the curve has no 2torsion, this is equal to the 2Selmer rank. If the curve has 2torsion, the upper bound may be smaller than the bound obtained from the 2Selmer rank minus the 2rank of the torsion, since more information is gained from the 2isogenous curve or curves.
EXAMPLES:
The following is the curve 960D1, which has rank 0, but Sha of order 4:
sage: E = mwrank_EllipticCurve([0, 1, 0, 900, 10098]) sage: E.rank_bound() 0 sage: E.rank() 0
In this case the rank was computed using a second descent, which is able to determine (by considering a 2isogenous curve) that Sha is nontrivial. If we deliberately stop the second descent, the rank bound is larger:
sage: E = mwrank_EllipticCurve([0, 1, 0, 900, 10098]) sage: E.two_descent(second_descent = False, verbose=False) sage: E.rank_bound() 2
In contrast, for the curve 571A, also with rank 0 and Sha of order 4, we only obtain an upper bound of 2:
sage: E = mwrank_EllipticCurve([0, 1, 1, 929, 10595]) sage: E.rank_bound() 2
In this case the value returned by
rank()
is only a lower bound in general (though this is correct):sage: E.rank() 0 sage: E.certain() False

regulator
()¶ Return the regulator of the saturated MordellWeil group.
EXAMPLES:
sage: E = mwrank_EllipticCurve([0, 0, 1, 1, 0]) sage: E.regulator() 0.05111140823996884

saturate
(bound=1)¶ Compute the saturation of the MordellWeil group at all primes up to
bound
.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 \(n\).
EXAMPLES:
Since the 2descent automatically saturates at primes up to 20, it is not easy to come up with an example where saturation has any effect:
sage: E = mwrank_EllipticCurve([0, 0, 0, 1002231243161, 0]) sage: E.gens() [[1001107, 4004428, 1]] sage: E.saturate() sage: E.gens() [[1001107, 4004428, 1]]
Check that trac ticket #18031 is fixed:
sage: E = EllipticCurve([0,1,1,266,968]) sage: Q1 = E([1995,3674,125]) sage: Q2 = E([157,1950,1]) sage: E.saturation([Q1,Q2]) ([(1 : 27 : 1), (157 : 1950 : 1)], 3, 0.801588644684981)

selmer_rank
()¶ Returns the rank of the 2Selmer group of the curve.
EXAMPLES:
The following is the curve 960D1, which has rank 0, but Sha of order 4. The 2torsion has rank 2, and the Selmer rank is 3:
sage: E = mwrank_EllipticCurve([0, 1, 0, 900, 10098]) sage: E.selmer_rank() 3
Nevertheless, we can obtain a tight upper bound on the rank since a second descent is performed which establishes the 2rank of Sha:
sage: E.rank_bound() 0
To show that this was resolved using a second descent, we do the computation again but turn off
second_descent
:sage: E = mwrank_EllipticCurve([0, 1, 0, 900, 10098]) sage: E.two_descent(second_descent = False, verbose=False) sage: E.rank_bound() 2
For the curve 571A, also with rank 0 and Sha of order 4, but with no 2torsion, the Selmer rank is strictly greater than the rank:
sage: E = mwrank_EllipticCurve([0, 1, 1, 929, 10595]) sage: E.selmer_rank() 2 sage: E.rank_bound() 2
In cases like this with no 2torsion, the rank upper bound is always equal to the 2Selmer rank. If we ask for the rank, all we get is a lower bound:
sage: E.rank() 0 sage: E.certain() False

set_verbose
(verbose)¶ Set the verbosity of printing of output by the
two_descent()
and other functions.INPUT:
verbose
(int) – if positive, print lots of output when doing 2descent.
EXAMPLES:
sage: E = mwrank_EllipticCurve([0, 0, 1, 1, 0]) sage: E.saturate() # no output sage: E.gens() [[0, 1, 1]] sage: E = mwrank_EllipticCurve([0, 0, 1, 1, 0]) sage: E.set_verbose(1) sage: E.saturate() # tol 1e10 Basic pair: I=48, J=432 disc=255744 2adic index bound = 2 By Lemma 5.1(a), 2adic index = 1 2adic index = 1 One (I,J) pair Looking for quartics with I = 48, J = 432 Looking for Type 2 quartics: Trying positive a from 1 up to 1 (square a first...) (1,0,6,4,1) trivial Trying positive a from 1 up to 1 (...then nonsquare a) Finished looking for Type 2 quartics. Looking for Type 1 quartics: Trying positive a from 1 up to 2 (square a first...) (1,0,0,4,4) nontrivial...(x:y:z) = (1 : 1 : 0) Point = [0:0:1] height = 0.0511114082399688402358 Rank of B=im(eps) increases to 1 (The previous point is on the egg) Exiting search for Type 1 quartics after finding one which is globally soluble. Mordell rank contribution from B=im(eps) = 1 Selmer rank contribution from B=im(eps) = 1 Sha rank contribution from B=im(eps) = 0 Mordell rank contribution from A=ker(eps) = 0 Selmer rank contribution from A=ker(eps) = 0 Sha rank contribution from A=ker(eps) = 0 Searching for points (bound = 8)...done: found points which generate a subgroup of rank 1 and regulator 0.0511114082399688402358 Processing points found during 2descent...done: now regulator = 0.0511114082399688402358 Saturating (with bound = 1)...done: points were already saturated.

silverman_bound
()¶ Return the Silverman height bound. This is a floating point number \(B\) such that if \(P\) is a point on the curve, then the naive logarithmic height \(h(P)\) is less than \(B+\hat{h}(P)\), where \(\hat{h}(P)\) is the canonical height of \(P\).
Warning
We assume the model is minimal!
EXAMPLES:
sage: E = mwrank_EllipticCurve([0, 0, 0, 1002231243161, 0]) sage: E.silverman_bound() 18.29545210468247 sage: E = mwrank_EllipticCurve([0,0,1,7,6]) sage: E.silverman_bound() 6.284833369972403

two_descent
(verbose=True, selmer_only=False, first_limit=20, second_limit=8, n_aux=1, second_descent=True)¶ Compute 2descent data for this curve.
INPUT:
verbose
(bool, defaultTrue
) – print what mwrank is doing.selmer_only
(bool, defaultFalse
) –selmer_only
switch.first_limit
(int, default 20) – bound on \(x+z\) in quartic point search.second_limit
(int, default 8) – bound on \(\log \max(x,z)\), i.e. logarithmic.n_aux
(int, default 1) – (only relevant for general 2descent when 2torsion trivial) number of primes used for quartic search.n_aux=1
causes default (8) to be used. Increase for curves of higher rank.second_descent
(bool, defaultTrue
) – (only relevant for curves with 2torsion, where mwrank uses descent via 2isogeny) flag determining whether or not to do second descent. Default strongly recommended.
OUTPUT:
Nothing – nothing is returned.

class
sage.libs.eclib.interface.
mwrank_MordellWeil
(curve, verbose=True, pp=1, maxr=999)¶ Bases:
sage.structure.sage_object.SageObject
The
mwrank_MordellWeil
class represents a subgroup of a MordellWeil group. Use this class to saturate a specified list of points on anmwrank_EllipticCurve
, or to search for points up to some bound.INPUT:
curve
(mwrank_EllipticCurve
) – the underlying elliptic curve.verbose
(bool, defaultFalse
) – verbosity flag (controls amount of output produced in point searches).pp
(int, default 1) – process points flag (if nonzero, the points found are processed, so that at all times only a \(\ZZ\)basis for the subgroup generated by the points found so far is stored; if zero, no processing is done and all points found are stored).maxr
(int, default 999) – maximum rank (quit point searching once the points found generate a subgroup of this rank; useful if an upper bound for the rank is already known).
EXAMPLES:
sage: E = mwrank_EllipticCurve([1,0,1,4,6]) sage: EQ = mwrank_MordellWeil(E) sage: EQ Subgroup of MordellWeil group: [] sage: EQ.search(2) P1 = [0:1:0] is torsion point, order 1 P1 = [1:1:1] is torsion point, order 2 P1 = [2:2:1] is torsion point, order 3 P1 = [9:23:1] is torsion point, order 6 sage: E = mwrank_EllipticCurve([0,0,1,7,6]) sage: EQ = mwrank_MordellWeil(E) sage: EQ.search(2) P1 = [0:1:0] is torsion point, order 1 P1 = [3:0:1] is generator number 1 ... P4 = [91:804:343] = 2*P1 + 2*P2 + 1*P3 (mod torsion) sage: EQ Subgroup of MordellWeil group: [[1:1:1], [2:3:1], [14:25:8]]
Example to illustrate the verbose parameter:
sage: E = mwrank_EllipticCurve([0,0,1,7,6]) sage: EQ = mwrank_MordellWeil(E, verbose=False) sage: EQ.search(1) sage: EQ Subgroup of MordellWeil group: [[1:1:1], [2:3:1], [14:25:8]] sage: EQ = mwrank_MordellWeil(E, verbose=True) 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 2saturation Points have successfully been 2saturated (max q used = 7) Checking 3saturation Points have successfully been 3saturated (max q used = 7) Checking 5saturation Points have successfully been 5saturated (max q used = 23) Checking 7saturation Points have successfully been 7saturated (max q used = 41) Checking 11saturation Points have successfully been 11saturated (max q used = 17) Checking 13saturation Points have successfully been 13saturated (max q used = 43) Checking 17saturation Points have successfully been 17saturated (max q used = 31) Checking 19saturation Points have successfully been 19saturated (max q used = 37) done P2 = [2:3:1] is generator number 2 saturating up to 20...Checking 2saturation possible kernel vector = [1,1] This point may be in 2E(Q): [14:52:1] ...and it is! Replacing old generator #1 with new generator [1:1:1] Points have successfully been 2saturated (max q used = 7) Index gain = 2^1 Checking 3saturation Points have successfully been 3saturated (max q used = 13) Checking 5saturation Points have successfully been 5saturated (max q used = 67) Checking 7saturation Points have successfully been 7saturated (max q used = 53) Checking 11saturation Points have successfully been 11saturated (max q used = 73) Checking 13saturation Points have successfully been 13saturated (max q used = 103) Checking 17saturation Points have successfully been 17saturated (max q used = 113) Checking 19saturation Points have successfully been 19saturated (max q used = 47) 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 2saturation Points have successfully been 2saturated (max q used = 11) Checking 3saturation Points have successfully been 3saturated (max q used = 13) Checking 5saturation Points have successfully been 5saturated (max q used = 71) Checking 7saturation Points have successfully been 7saturated (max q used = 101) Checking 11saturation Points have successfully been 11saturated (max q used = 127) Checking 13saturation Points have successfully been 13saturated (max q used = 151) Checking 17saturation Points have successfully been 17saturated (max q used = 139) Checking 19saturation Points have successfully been 19saturated (max q used = 179) 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) P4 = [1:0:1] = 1*P1 + 0*P2 + 0*P3 (mod torsion) P4 = [2:0:1] = 1*P1 + 1*P2 + 0*P3 (mod torsion) P4 = [18:7:8] = 2*P1 + 1*P2 + 1*P3 (mod torsion) P4 = [3:3:1] = 1*P1 + 0*P2 + 1*P3 (mod torsion) P4 = [4:6:1] = 0*P1 + 1*P2 + 1*P3 (mod torsion) P4 = [36:69:64] = 1*P1 + 2*P2 + 0*P3 (mod torsion) P4 = [68:25:64] = 2*P1 + 1*P2 + 2*P3 (mod torsion) P4 = [12:35:27] = 1*P1 + 1*P2 + 1*P3 (mod torsion) sage: EQ Subgroup of MordellWeil group: [[1:1:1], [2:3:1], [14:25:8]]
Example to illustrate the process points (
pp
) parameter:sage: E = mwrank_EllipticCurve([0,0,1,7,6]) sage: EQ = mwrank_MordellWeil(E, verbose=False, pp=1) sage: EQ.search(1); EQ # generators only Subgroup of MordellWeil group: [[1:1:1], [2:3:1], [14:25:8]] sage: EQ = mwrank_MordellWeil(E, verbose=False, pp=0) sage: EQ.search(1); EQ # all points found Subgroup of MordellWeil group: [[3:0:1], [2:3:1], [14:25:8], [1:3:1], [0:2:1], [2:13:8], [1:0:1], [2:0:1], [18:7:8], [3:3:1], [4:6:1], [36:69:64], [68:25:64], [12:35:27]]

points
()¶ Return a list of the generating points in this MordellWeil group.
OUTPUT:
(list) A list of lists of length 3, each holding the primitive integer coordinates \([x,y,z]\) of a generating point.
EXAMPLES:
sage: E = mwrank_EllipticCurve([0,0,1,7,6]) sage: EQ = mwrank_MordellWeil(E) 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) sage: EQ.points() [[1, 1, 1], [2, 3, 1], [14, 25, 8]]

process
(v, sat=0)¶ This function allows one to add points to a
mwrank_MordellWeil
object.Process points in the list
v
, with saturation at primes up tosat
. Ifsat
is zero (the default), do no saturation.INPUT:
v
(list of 3tuples 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 tosat
, or at all primes ifsat
is zero.
OUTPUT:
None. But note that if the
verbose
flag is set, then there will be some output as a sideeffect.EXAMPLES:
sage: E = mwrank_EllipticCurve([0,0,1,7,6]) sage: E.gens() [[1, 1, 1], [2, 3, 1], [14, 25, 8]] sage: EQ = mwrank_MordellWeil(E) sage: EQ.process([[1, 1, 1], [2, 3, 1], [14, 25, 8]]) P1 = [1:1:1] is generator number 1 P2 = [2:3:1] is generator number 2 P3 = [14:25:8] is generator number 3
sage: EQ.points() [[1, 1, 1], [2, 3, 1], [14, 25, 8]]
Example to illustrate the saturation parameter
sat
: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) P1 = [1547:2967:343] is generator number 1 ... Gained index 5, new generators = [ [2:3:1] [14:25:8] [1:1:1] ] sage: EQ.points() [[2, 3, 1], [14, 25, 8], [1, 1, 1]]
Here the processing was followed by saturation at primes up to 20. Now we prevent this initial saturation:
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) P1 = [1547:2967:343] is generator number 1 P2 = [2707496766203306:864581029138191:2969715140223272] is generator number 2 P3 = [13422227300:49322830557:12167000000] is generator number 3 sage: EQ.points() [[1547, 2967, 343], [2707496766203306, 864581029138191, 2969715140223272], [13422227300, 49322830557, 12167000000]] sage: EQ.regulator() 375.42920288254555 sage: EQ.saturate(2) # points were not 2saturated saturating basis...Saturation index bound = 93 WARNING: saturation at primes p > 2 will not be done; ... Gained index 2 New regulator = 93.857... (False, 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 3saturated saturating basis...Saturation index bound = 46 WARNING: saturation at primes p > 3 will not be done; ... Gained index 3 New regulator = 10.428... (False, 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 5saturated saturating basis...Saturation index bound = 15 WARNING: saturation at primes p > 5 will not be done; ... Gained index 5 New regulator = 0.417... (False, 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 Checking saturation at [ 2 3 ] Checking 2saturation Points were proved 2saturated (max q used = 11) Checking 3saturation Points were proved 3saturated (max q used = 13) done (True, 1, '[ ]')

rank
()¶ Return the rank of this subgroup of the MordellWeil group.
OUTPUT:
(int) The rank of this subgroup of the MordellWeil group.
EXAMPLES:
sage: E = mwrank_EllipticCurve([0,1,1,0,0]) sage: E.rank() 0
A rank 3 example:
sage: E = mwrank_EllipticCurve([0,0,1,7,6]) sage: EQ = mwrank_MordellWeil(E) sage: EQ.rank() 0 sage: EQ.regulator() 1.0
The preceding output is correct, since we have not yet tried to find any points on the curve either by searching or 2descent:
sage: EQ Subgroup of MordellWeil group: []
Now we do a very small search:
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 2saturation ... P4 = [12:35:27] = 1*P1 + 1*P2 + 1*P3 (mod torsion) sage: EQ Subgroup of MordellWeil group: [[1:1:1], [2:3:1], [14:25:8]] sage: EQ.rank() 3 sage: EQ.regulator() 0.417143558758384
We do in fact now have a full MordellWeil basis.

regulator
()¶ Return the regulator of the points in this subgroup of the MordellWeil group.
Note
eclib
can compute the regulator to arbitrary precision, but the interface currently returns the output as afloat
.OUTPUT:
(float) The regulator of the points in this subgroup.
EXAMPLES:
sage: E = mwrank_EllipticCurve([0,1,1,0,0]) sage: E.regulator() 1.0 sage: E = mwrank_EllipticCurve([0,0,1,7,6]) sage: E.regulator() 0.417143558758384

saturate
(max_prime=1, odd_primes_only=False)¶ Saturate this subgroup of the MordellWeil group.
INPUT:
max_prime
(int, default 1) – saturation is performed for all primes up tomax_prime
. 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 theeclib
library (currently 97) then no saturation will be attempted at primes above this.odd_primes_only
(bool, defaultFalse
) – only do saturation at odd primes. (If the points have been found viatwo_descent()
they should already be 2saturated.)
OUTPUT:
(3tuple) (
ok
,index
,unsatlist
) where:ok
(bool) –True
if and only if the saturation was provably successful at all primes attempted. If the default was used formax_prime
and no warning was output about the computed saturation bound being too high, thenTrue
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 thateclib
uses floating point methods based on elliptic logarithms to divide points.)
Note
We emphasize that if this function returns
True
as the first return argument (ok
), and if the default was used for the parametermax_prime
, then the points in the basis after calling this function are saturated at all primes, i.e., saturating at the primes up tomax_prime
are sufficient to saturate at all primes. Note that the function might not have needed to saturate at all primes up tomax_prime
. It has worked out what prime you need to saturate up to, and that prime might be smaller thanmax_prime
.Note
Currently (May 2010), this does not remember the result of calling
search()
. So callingsearch()
up to height 20 then callingsaturate()
results in another search up to height 18.EXAMPLES:
sage: E = mwrank_EllipticCurve([0,0,1,7,6]) sage: EQ = mwrank_MordellWeil(E)
We initialise with three points which happen to be 2, 3 and 5 times the generators of this rank 3 curve. To prevent 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) P1 = [1547:2967:343] is generator number 1 P2 = [2707496766203306:864581029138191:2969715140223272] is generator number 2 P3 = [13422227300:49322830557:12167000000] is generator number 3 sage: EQ Subgroup of MordellWeil group: [[1547:2967:343], [2707496766203306:864581029138191:2969715140223272], [13422227300:49322830557:12167000000]] sage: EQ.regulator() 375.42920288254555
Now we saturate at \(p=2\), and gain index 2:
sage: EQ.saturate(2) # points were not 2saturated saturating basis...Saturation index bound = 93 WARNING: saturation at primes p > 2 will not be done; ... Gained index 2 New regulator = 93.857... (False, 2, '[ ]') sage: EQ Subgroup of MordellWeil group: [[2:3:1], [2707496766203306:864581029138191:2969715140223272], [13422227300:49322830557:12167000000]] sage: EQ.regulator() 93.85730072063639
Now we saturate at \(p=3\), and gain index 3:
sage: EQ.saturate(3) # points were not 3saturated saturating basis...Saturation index bound = 46 WARNING: saturation at primes p > 3 will not be done; ... Gained index 3 New regulator = 10.428... (False, 3, '[ ]') sage: EQ Subgroup of MordellWeil group: [[2:3:1], [14:25:8], [13422227300:49322830557:12167000000]] sage: EQ.regulator() 10.4285889689596
Now we saturate at \(p=5\), and gain index 5:
sage: EQ.saturate(5) # points were not 5saturated saturating basis...Saturation index bound = 15 WARNING: saturation at primes p > 5 will not be done; ... Gained index 5 New regulator = 0.417... (False, 5, '[ ]') sage: EQ Subgroup of MordellWeil group: [[2:3:1], [14:25:8], [1:1:1]] sage: EQ.regulator() 0.417143558758384
Finally we finish the saturation. The output here shows that the points are now provably saturated at all primes:
sage: EQ.saturate() # points are now saturated saturating basis...Saturation index bound = 3 Checking saturation at [ 2 3 ] Checking 2saturation Points were proved 2saturated (max q used = 11) Checking 3saturation Points were proved 3saturated (max q used = 13) done (True, 1, '[ ]')
Of course, the
process()
function would have done all this automatically for us: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) P1 = [1547:2967:343] is generator number 1 ... Gained index 5, new generators = [ [2:3:1] [14:25:8] [1:1:1] ] sage: EQ Subgroup of MordellWeil group: [[2:3:1], [14:25:8], [1:1:1]] sage: EQ.regulator() 0.417143558758384
But we would still need to use the
saturate()
function to verify that full saturation has been done:sage: EQ.saturate() saturating basis...Saturation index bound = 3 Checking saturation at [ 2 3 ] Checking 2saturation Points were proved 2saturated (max q used = 11) Checking 3saturation Points were proved 3saturated (max q used = 13) done (True, 1, '[ ]')
Note the output of the preceding command: it proves that the 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.

search
(height_limit=18, verbose=False)¶ Search for new points, and add them to this subgroup of the MordellWeil group.
INPUT:
height_limit
(float, default: 18) – search up to this logarithmic height.
Note
On 32bit machines, this must be < 21.48 else \(\exp(h_{\text{lim}}) > 2^{31}\) and overflows. On 64bit machines, it must be at most 43.668. 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.
Note
The search is carried out with a quadratic sieve, using code adapted from a version of Michael Stoll’s
ratpoints
program. It would be preferable to use a newer version ofratpoints
.verbose
(bool, defaultFalse
) – turn verbose operation on or off.
EXAMPLES:
A rank 3 example, where a very small search is sufficient to find a MordellWeil basis:
sage: E = mwrank_EllipticCurve([0,0,1,7,6]) sage: EQ = mwrank_MordellWeil(E) 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) sage: EQ Subgroup of MordellWeil group: [[1:1:1], [2:3:1], [14:25:8]]
In the next example, a search bound of 12 is needed to find a nontorsion point:
sage: E = mwrank_EllipticCurve([0, 1, 0, 18392, 1186248]) #1056g4 sage: EQ = mwrank_MordellWeil(E) sage: EQ.search(11); EQ P1 = [0:1:0] is torsion point, order 1 P1 = [161:0:1] is torsion point, order 2 Subgroup of MordellWeil group: [] sage: EQ.search(12); EQ P1 = [0:1:0] is torsion point, order 1 P1 = [161:0:1] is torsion point, order 2 P1 = [4413270:10381877:27000] is generator number 1 ... Subgroup of MordellWeil group: [[4413270:10381877:27000]]