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# elliptic curve utility functions for finding generators, saturating and mapping around an isogeny class
from sage.all import (pari, QQ,
mwrank_get_precision, mwrank_set_precision)
from magma import get_magma, MagmaEffort
mwrank_saturation_precision = 1000 # 500 not enough for 594594bf2
mwrank_saturation_maxprime = 1000
GP = '/usr/local/bin/gp'
# Assuming that E is known to have rank 1, returns a point on E
# computed by Magma's HeegnerPoint command
def magma_rank1_gen(E, mE):
mP = mE.HeegnerPoint(nvals=2)[1]
P = E([mP[i].sage() for i in [1, 2, 3]])
return P
# Assuming that E is known to have rank 1, returns a point on E
# computed by GP's ellheegner() command
def pari_rank1_gen_old(E, stacksize=1024000000):
from os import system, getpid, unlink
f = 'tempfile-'+str(getpid())
comm = "LD_LIBRARY_PATH=/usr/local/lib; echo `echo 'ellheegner(ellinit("+str(list(E.ainvs()))+"))' | %s -q -f -s %s` > %s;" % (GP, stacksize, f)
system(comm)
P = open(f).read()
#print(P)
P = open(f).read().partition("[")[2].partition("]")[0]
P = P.replace("\xb1", "") # needed for 497805u1
#print(P)
unlink(f)
P = E([QQ(c) for c in P.split(',')])
#print(P)
return P
def pari_rank1_gen(E):
return E(pari(E).ellheegner().sage())
def get_magma_gens(E, mE):
MS = mE.MordellWeilShaInformation(RankOnly=True, Effort=MagmaEffort, nvals=3)
rank_bounds = [r.sage() for r in MS[0]]
gens = [E(P.Eltseq().sage()) for P in MS[1]]
return rank_bounds, gens
def get_gens_mwrank(E):
return E.gens(algorithm='mwrank_lib', descent_second_limit=15, sat_bound=2)
def get_rank1_gens(E, mE, verbose=0):
if verbose:
print(" - trying a point search...")
gens = E.point_search(15)
if gens:
if verbose:
print("--success: P = {}".format(gens[0]))
return gens
if verbose:
print("--failed. Trying pari's ellheegner...")
gens = [pari_rank1_gen(E)]
if gens:
if verbose:
print("--success: P = {}".format(gens[0]))
return gens
if verbose:
print("--failed. Trying Magma's HeegnerPoint...")
try:
gens = [magma_rank1_gen(E, mE)]
if gens:
if verbose:
print("--success: P = {}".format(gens[0]))
return gens
except:
pass
if verbose:
print("-- failed. Trying Magma...")
_, gens = get_magma_gens(E, mE)
if gens:
if verbose:
print("--success: P = {}".format(gens[0]))
return gens
if verbose:
print("--failed. Trying mwrank...")
return get_gens_mwrank(E)
def get_gens_simon(E):
E.simon_two_descent(lim3=5000)
return E.gens()
def get_gens(E, ar, verbose=0):
if ar == 0:
return []
mag = get_magma()
mE = mag(E)
if ar == 1:
if verbose > 1:
print("{}: a.r.=1, finding a generator".format(E.ainvs()))
gens = get_rank1_gens(E, mE, verbose)
else: # ar >=2
if verbose > 1:
print("{}: a.r.={}, finding generators using Magma".format(E.ainvs(), ar))
_, gens = get_magma_gens(E, mE)
if verbose > 1:
print("gens = {}".format(gens))
# Now we have independent gens, and saturate them
prec0 = mwrank_get_precision()
mwrank_set_precision(mwrank_saturation_precision)
if verbose > 1:
print("Starting saturation (automatic saturation bound)...")
gens, index, reg = E.saturation(gens, max_prime=-1)
# if verbose > 1:
# print("Starting saturation (p<{})...".format(mwrank_saturation_maxprime))
# gens, index, reg = E.saturation(gens, max_prime=mwrank_saturation_maxprime)
mwrank_set_precision(prec0)
if verbose > 1:
print("... finished saturation (index {}, new reg={})".format(index, reg))
return gens
# Given a matrix of isogenies and a list of points on the initial
# curve returns a# list of their images on each other curve. The
# complication is that the isogenies will only exist when they have
# prime degree.
# Here we assume that the points in Plist are saturated, and only
# resaturate their images at primes up to the maximum prime dividing
# an isogeny degree.
def map_points(maps, Plist, verbose=0):
ncurves = len(maps)
if len(Plist) == 0:
return [[] for _ in range(ncurves)]
if ncurves == 1:
return [Plist]
if verbose > 1:
print("in map_points with degrees {}".format([[phi.degree() if phi else 0 for phi in r] for r in maps]))
maxp = max([max([max(phi.degree().support(), default=0) if phi else 0 for phi in r], default=0) for r in maps], default=0)
if verbose > 1:
print(" maxp = {}".format(maxp))
Qlists = [Plist] + [[]]*(ncurves-1)
nfill = 1
for i in range(ncurves):
if nfill == ncurves:
break
for j in range(1, ncurves):
if (maps[i][j] != 0) and Qlists[j] == []:
if verbose>1:
print("Mapping points from curve {} to curve {} via {}".format(i,j,maps[i][j]))
print("points to be mapped: {}".format(Qlists[i]))
Qlists[j] = [maps[i][j](P) for P in Qlists[i]]
nfill += 1
# now we saturate the points just computed at all primes up to maxp
prec0 = mwrank_get_precision()
mwrank_set_precision(mwrank_saturation_precision)
for i in range(1, ncurves):
E = Qlists[i][0].curve()
if verbose > 1:
print("Saturating curve {} (maxp={})...".format(i, maxp))
Qlists[i], n, _ = E.saturation(Qlists[i], max_prime=maxp)
if verbose > 1:
print("--saturation index was {}".format(n))
mwrank_set_precision(prec0)
return Qlists
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