def _repr_(self):
"""
Return string representation of self.
EXAMPLES::
sage: A.<x,y,z>=FreeAlgebra(ZZ,3)
sage: repr(-x+3*y*z) # indirect doctest
'-x + 3*y*z'
Trac ticket :trac:`11068` enables the use of local variable names::
sage: from sage.structure.parent_gens import localvars
sage: with localvars(A, ['a','b','c']):
....: print(-x+3*y*z)
-a + 3*b*c
"""
v = sorted(self._monomial_coefficients.items())
P = self.parent()
M = P.monoid()
from sage.structure.parent_gens import localvars
with localvars(M, P.variable_names(), normalize=False):
x = repr_lincomb(v, strip_one=True)
return x
def _repr_(self):
"""
Return string representation of self.
EXAMPLES::
sage: A.<x,y,z>=FreeAlgebra(ZZ,3)
sage: repr(-x+3*y*z)
'-x + 3*y*z'
Trac ticket #11068 enables the use of local variable names::
sage: from sage.structure.parent_gens import localvars
sage: with localvars(A, ['a','b','c']):
... print -x+3*y*z
...
-a + 3*b*c
"""
v = self.__monomial_coefficients.items()
v.sort()
mons = [ m for (m, _) in v ]
cffs = [ x for (_, x) in v ]
P = self.parent()
M = P.monoid()
from sage.structure.parent_gens import localvars
with localvars(M, P.variable_names(), normalize=False):
x = repr_lincomb(mons, cffs).replace("*1 "," ")
if x[len(x)-2:] == "*1":
return x[:len(x)-2]
else:
return x
def _repr_(self):
"""
Return string representation of self.
EXAMPLES::
sage: A.<x,y,z>=FreeAlgebra(ZZ,3)
sage: repr(-x+3*y*z)
'-x + 3*y*z'
Trac ticket #11068 enables the use of local variable names::
sage: from sage.structure.parent_gens import localvars
sage: with localvars(A, ['a','b','c']):
... print -x+3*y*z
...
-a + 3*b*c
"""
v = self.__monomial_coefficients.items()
v.sort()
mons = [m for (m, _) in v]
cffs = [x for (_, x) in v]
P = self.parent()
M = P.monoid()
from sage.structure.parent_gens import localvars
with localvars(M, P.variable_names(), normalize=False):
x = repr_lincomb(mons, cffs).replace("*1 ", " ")
if x[len(x) - 2:] == "*1":
return x[:len(x) - 2]
else:
return x
def _repr_(self):
"""
Return string representation of self.
EXAMPLES::
sage: A.<x,y,z>=FreeAlgebra(ZZ,3)
sage: repr(-x+3*y*z) # indirect doctest
'-x + 3*y*z'
Trac ticket :trac:`11068` enables the use of local variable names::
sage: from sage.structure.parent_gens import localvars
sage: with localvars(A, ['a','b','c']):
....: print(-x+3*y*z)
-a + 3*b*c
"""
v = sorted(self._monomial_coefficients.items())
P = self.parent()
M = P.monoid()
from sage.structure.parent_gens import localvars
with localvars(M, P.variable_names(), normalize=False):
x = repr_lincomb(v, strip_one=True)
return x
def _latex_(self):
"""
EXAMPLES::
sage: H, (i,j,k) = sage.algebras.free_algebra_quotient.hamilton_quatalg(QQ)
sage: ((2/3)*i - j)._latex_()
'\\frac{2}{3}i + \\left(-1\\right)j'
"""
Q = self.parent()
M = Q.monoid()
with localvars(M, Q.variable_names()):
cffs = tuple(self.__vector)
mons = Q.monomial_basis()
return repr_lincomb(zip(mons, cffs), is_latex=True, strip_one=True)
def _repr_(self):
"""
EXAMPLES::
sage: H, (i,j,k) = sage.algebras.free_algebra_quotient.hamilton_quatalg(ZZ)
sage: i._repr_()
'i'
"""
Q = self.parent()
M = Q.monoid()
with localvars(M, Q.variable_names()):
cffs = list(self.__vector)
mons = Q.monomial_basis()
return repr_lincomb(zip(mons, cffs), strip_one=True)
def _repr_(self):
"""
EXAMPLES::
sage: H, (i,j,k) = sage.algebras.free_algebra_quotient.hamilton_quatalg(ZZ)
sage: i._repr_()
'i'
"""
Q = self.parent()
M = Q.monoid()
with localvars(M, Q.variable_names()):
cffs = list(self.__vector)
mons = Q.monomial_basis()
return repr_lincomb(zip(mons, cffs), strip_one=True)
def _latex_(self):
"""
EXAMPLES::
sage: H, (i,j,k) = sage.algebras.free_algebra_quotient.hamilton_quatalg(QQ)
sage: ((2/3)*i - j)._latex_()
'\\frac{2}{3}i - j'
"""
Q = self.parent()
M = Q.monoid()
with localvars(M, Q.variable_names()):
cffs = tuple(self.__vector)
mons = Q.monomial_basis()
return repr_lincomb(zip(mons, cffs), is_latex=True, strip_one=True)
def _repr_(self):
"""
EXAMPLES::
sage: H, (i,j,k) = sage.algebras.free_algebra_quotient.hamilton_quatalg(ZZ)
sage: i._repr_()
'i'
"""
Q = self.parent()
M = Q.monoid()
with localvars(M, Q.variable_names()):
cffs = list(self.__vector)
mons = Q.monomial_basis()
x = repr_lincomb(mons, cffs).replace("*1 ", " ")
if x[len(x) - 2:] == "*1":
return x[:len(x) - 2]
else:
return x
def _latex_(self):
"""
EXAMPLES::
sage: H, (i,j,k) = sage.algebras.free_algebra_quotient.hamilton_quatalg(QQ)
sage: ((2/3)*i - j)._latex_()
'\\frac{2}{3}i + \\left(-1\\right)j'
"""
Q = self.parent()
M = Q.monoid()
with localvars(M, Q.variable_names()):
cffs = list(self.__vector)
mons = Q.monomial_basis()
x = repr_lincomb(mons, cffs, True).replace("*1 "," ")
if x[len(x)-2:] == "*1":
return x[:len(x)-2]
else:
return x
def _latex_(self):
"""
EXAMPLES::
sage: H, (i,j,k) = sage.algebras.free_algebra_quotient.hamilton_quatalg(QQ)
sage: ((2/3)*i - j)._latex_()
'\\frac{2}{3}i + \\left(-1\\right)j'
"""
Q = self.parent()
M = Q.monoid()
with localvars(M, Q.variable_names()):
cffs = list(self.__vector)
mons = Q.monomial_basis()
x = repr_lincomb(mons, cffs, True).replace("*1 ", " ")
if x[len(x) - 2:] == "*1":
return x[:len(x) - 2]
else:
return x
def _repr_(self):
"""
EXAMPLES::
sage: H, (i,j,k) = sage.algebras.free_algebra_quotient.hamilton_quatalg(ZZ)
sage: i._repr_()
'i'
"""
Q = self.parent()
M = Q.monoid()
with localvars(M, Q.variable_names()):
cffs = list(self.__vector)
mons = Q.monomial_basis()
x = repr_lincomb(mons, cffs).replace("*1 "," ")
if x[len(x)-2:] == "*1":
return x[:len(x)-2]
else:
return x
def _repr_(self):
"""
String representation.
TESTS::
sage: R.<x,y> = QQ[]; S.<a,b> = R.quo(x^2 + y^2); type(a)
<class 'sage.rings.quotient_ring_element.QuotientRing_generic_with_category.element_class'>
sage: a-2*a*b # indirect doctest
-2*a*b + a
In trac ticket #11068, the case of quotient rings without
assigned names has been covered as well::
sage: S = SteenrodAlgebra(2)
sage: I = S*[S.0+S.1]*S
sage: Q = S.quo(I)
sage: Q.0
Sq(1)
"""
from sage.structure.parent_gens import localvars
P = self.parent()
R = P.cover_ring()
# We print by temporarily (and safely!) changing the variable
# names of the covering structure R to those of P.
# These names get changed back, since we're using "with".
# However, it may occur that no variable names are assigned.
# That holds, in particular, if there are infinitely many
# generators, as for Steenrod algebras.
try:
names = P.variable_names()
except ValueError:
return str(self.__rep)
with localvars(R, P.variable_names(), normalize=False):
return str(self.__rep)
def _repr_(self):
"""
String representation.
TESTS::
sage: R.<x,y> = QQ[]; S.<a,b> = R.quo(x^2 + y^2); type(a)
<class 'sage.rings.quotient_ring_element.QuotientRing_generic_with_category.element_class'>
sage: a-2*a*b # indirect doctest
-2*a*b + a
In trac ticket #11068, the case of quotient rings without
assigned names has been covered as well::
sage: S = SteenrodAlgebra(2)
sage: I = S*[S.0+S.1]*S
sage: Q = S.quo(I)
sage: Q.0
Sq(1)
"""
from sage.structure.parent_gens import localvars
P = self.parent()
R = P.cover_ring()
# We print by temporarily (and safely!) changing the variable
# names of the covering structure R to those of P.
# These names get changed back, since we're using "with".
# However, it may occur that no variable names are assigned.
# That holds, in particular, if there are infinitely many
# generators, as for Steenrod algebras.
try:
names = P.variable_names()
except ValueError:
return str(self.__rep)
with localvars(R, P.variable_names(), normalize=False):
return str(self.__rep)
def simon_two_descent(E,
verbose=0,
lim1=None,
lim3=None,
limtriv=None,
maxprob=20,
limbigprime=30,
known_points=[]):
"""
Interface to Simon's gp script for two-descent.
.. NOTE::
Users should instead run E.simon_two_descent()
EXAMPLES::
sage: import sage.schemes.elliptic_curves.gp_simon
sage: E=EllipticCurve('389a1')
sage: sage.schemes.elliptic_curves.gp_simon.simon_two_descent(E)
(2, 2, [(5/4 : 5/8 : 1), (-3/4 : 7/8 : 1)])
TESTS::
sage: E = EllipticCurve('37a1').change_ring(QuadraticField(-11,'x'))
sage: E.simon_two_descent()
(1, 1, [(0 : 0 : 1)])
An example with an elliptic curve defined over a relative number field::
sage: F.<a> = QuadraticField(29)
sage: x = QQ['x'].gen()
sage: K.<b> = F.extension(x^2-1/2*a+1/2)
sage: E = EllipticCurve(K,[1, 0, 5/2*a + 27/2, 0, 0]) # long time (about 3 s)
sage: E.simon_two_descent(lim1=2, limtriv=3)
(1, 1, ...)
Check that :trac:`16022` is fixed::
sage: K.<y> = NumberField(x^4 + x^2 - 7)
sage: E = EllipticCurve(K, [1, 0, 5*y^2 + 16, 0, 0])
sage: E.simon_two_descent(lim1=2, limtriv=3) # long time (about 3 s)
(1, 1, ...)
An example that checks that :trac:`9322` is fixed (it should take less than a second to run)::
sage: K.<w> = NumberField(x^2-x-232)
sage: E = EllipticCurve([2-w,18+3*w,209+9*w,2581+175*w,852-55*w])
sage: E.simon_two_descent()
(0, 2, [])
"""
init()
current_randstate().set_seed_gp(gp)
K = E.base_ring()
K_orig = K
# The following is to correct the bug at \#5204: the gp script
# fails when K is a number field whose generator is called 'x'.
# It also deals with relative number fields.
E_orig = E
if not K is QQ:
K = K_orig.absolute_field('a')
from_K, to_K = K.structure()
E = E_orig.change_ring(to_K)
known_points = [P.change_ring(to_K) for P in known_points]
# Simon's program requires that this name be y.
with localvars(K.polynomial().parent(), 'y'):
gp.eval("K = bnfinit(%s);" % K.polynomial())
if verbose >= 2:
print("K = bnfinit(%s);" % K.polynomial())
gp.eval("%s = Mod(y,K.pol);" % K.gen())
if verbose >= 2:
print("%s = Mod(y,K.pol);" % K.gen())
else:
from_K = lambda x: x
to_K = lambda x: x
# The block below mimicks the defaults in Simon's scripts, and needs to be changed
# when these are updated.
if K is QQ:
cmd = 'ellrank(%s, %s);' % (list(
E.ainvs()), [P._pari_() for P in known_points])
if lim1 is None:
lim1 = 5
if lim3 is None:
lim3 = 50
if limtriv is None:
limtriv = 3
else:
cmd = 'bnfellrank(K, %s, %s);' % (list(
E.ainvs()), [P._pari_() for P in known_points])
if lim1 is None:
lim1 = 2
if lim3 is None:
lim3 = 4
if limtriv is None:
limtriv = 2
gp('DEBUGLEVEL_ell=%s; LIM1=%s; LIM3=%s; LIMTRIV=%s; MAXPROB=%s; LIMBIGPRIME=%s;'
% (verbose, lim1, lim3, limtriv, maxprob, limbigprime))
if verbose >= 2:
print(cmd)
s = gp.eval('ans=%s;' % cmd)
if s.find(" *** ") != -1:
raise RuntimeError(
"\n%s\nAn error occurred while running Simon's 2-descent program" %
s)
if verbose > 0:
print(s)
v = gp.eval('ans')
if v == 'ans': # then the call to ellrank() or bnfellrank() failed
raise RuntimeError(
"An error occurred while running Simon's 2-descent program")
if verbose >= 2:
print("v = %s" % v)
# pari represents field elements as Mod(poly, defining-poly)
# so this function will return the respective elements of K
def _gp_mod(*args):
return args[0]
ans = sage_eval(v, {'Mod': _gp_mod, 'y': K.gen(0)})
lower = ZZ(ans[0])
upper = ZZ(ans[1])
points = [E_orig([from_K(c) for c in list(P)]) for P in ans[2]]
points = [P for P in points if P.has_infinite_order()]
return lower, upper, points
def simon_two_descent(E, verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30):
"""
Interface to Simon's gp script for two-descent.
.. NOTE::
Users should instead run E.simon_two_descent()
EXAMPLES::
sage: import sage.schemes.elliptic_curves.gp_simon
sage: E=EllipticCurve('389a1')
sage: sage.schemes.elliptic_curves.gp_simon.simon_two_descent(E)
[2, 2, [(1 : 0 : 1), (-11/9 : -55/27 : 1)]]
sage: E.simon_two_descent()
(2, 2, [(1 : 0 : 1), (-11/9 : -55/27 : 1)])
TESTS::
sage: E = EllipticCurve('37a1').change_ring(QuadraticField(-11,'x'))
sage: E.simon_two_descent()
(1, 1, [(-1 : 0 : 1)])
"""
init()
current_randstate().set_seed_gp(gp)
K = E.base_ring()
K_orig = K
# The following is to correct the bug at \#5204: the gp script
# fails when K is a number field whose generator is called 'x'.
if not K is QQ:
K = K.change_names('a')
E_orig = E
E = EllipticCurve(K,[K(list(a)) for a in E.ainvs()])
F = E.integral_model()
if K != QQ:
# Simon's program requires that this name be y.
with localvars(K.polynomial().parent(), 'y'):
gp.eval("K = bnfinit(%s);" % K.polynomial())
if verbose >= 2:
print "K = bnfinit(%s);" % K.polynomial()
gp.eval("%s = Mod(y,K.pol);" % K.gen())
if verbose >= 2:
print "%s = Mod(y,K.pol);" % K.gen()
if K == QQ:
cmd = 'ellrank([%s,%s,%s,%s,%s]);' % F.ainvs()
else:
cmd = 'bnfellrank(K, [%s,%s,%s,%s,%s]);' % F.ainvs()
gp('DEBUGLEVEL_ell=%s; LIM1=%s; LIM3=%s; LIMTRIV=%s; MAXPROB=%s; LIMBIGPRIME=%s;'%(
verbose, lim1, lim3, limtriv, maxprob, limbigprime))
if verbose >= 2:
print cmd
s = gp.eval('ans=%s;'%cmd)
if s.find("***") != -1:
raise RuntimeError, "\n%s\nAn error occurred while running Simon's 2-descent program"%s
if verbose > 0:
print s
v = gp.eval('ans')
if v=='ans': # then the call to ellrank() or bnfellrank() failed
return 'fail'
if verbose >= 2:
print "v = ", v
# pari represents field elements as Mod(poly, defining-poly)
# so this function will return the respective elements of K
def _gp_mod(*args):
return args[0]
ans = sage_eval(v, {'Mod': _gp_mod, 'y': K.gen(0)})
inv_transform = F.isomorphism_to(E)
ans[2] = [inv_transform(F(P)) for P in ans[2]]
ans[2] = [E_orig([K_orig(list(c)) for c in list(P)]) for P in ans[2]]
return ans
def simon_two_descent(E, verbose=0, lim1=None, lim3=None, limtriv=None,
maxprob=20, limbigprime=30, known_points=[]):
"""
Interface to Simon's gp script for two-descent.
.. NOTE::
Users should instead run E.simon_two_descent()
EXAMPLES::
sage: import sage.schemes.elliptic_curves.gp_simon
sage: E=EllipticCurve('389a1')
sage: sage.schemes.elliptic_curves.gp_simon.simon_two_descent(E)
(2, 2, [(5/4 : 5/8 : 1), (-3/4 : 7/8 : 1)])
TESTS::
sage: E = EllipticCurve('37a1').change_ring(QuadraticField(-11,'x'))
sage: E.simon_two_descent()
(1, 1, [(0 : 0 : 1)])
An example with an elliptic curve defined over a relative number field::
sage: F.<a> = QuadraticField(29)
sage: x = QQ['x'].gen()
sage: K.<b> = F.extension(x^2-1/2*a+1/2)
sage: E = EllipticCurve(K,[1, 0, 5/2*a + 27/2, 0, 0]) # long time (about 3 s)
sage: E.simon_two_descent(lim1=2, limtriv=3)
(1, 1, ...)
Check that :trac:`16022` is fixed::
sage: K.<y> = NumberField(x^4 + x^2 - 7)
sage: E = EllipticCurve(K, [1, 0, 5*y^2 + 16, 0, 0])
sage: E.simon_two_descent(lim1=2, limtriv=3) # long time (about 3 s)
(1, 1, ...)
An example that checks that :trac:`9322` is fixed (it should take less than a second to run)
sage: K.<w> = NumberField(x^2-x-232)
sage: E = EllipticCurve([2-w,18+3*w,209+9*w,2581+175*w,852-55*w])
sage: E.simon_two_descent()
(0, 2, [])
"""
init()
current_randstate().set_seed_gp(gp)
K = E.base_ring()
K_orig = K
# The following is to correct the bug at \#5204: the gp script
# fails when K is a number field whose generator is called 'x'.
# It also deals with relative number fields.
E_orig = E
if not K is QQ:
K = K_orig.absolute_field('a')
from_K,to_K = K.structure()
E = E_orig.change_ring(to_K)
known_points = [P.change_ring(to_K) for P in known_points]
# Simon's program requires that this name be y.
with localvars(K.polynomial().parent(), 'y'):
gp.eval("K = bnfinit(%s);" % K.polynomial())
if verbose >= 2:
print("K = bnfinit(%s);" % K.polynomial())
gp.eval("%s = Mod(y,K.pol);" % K.gen())
if verbose >= 2:
print("%s = Mod(y,K.pol);" % K.gen())
else:
from_K = lambda x: x
to_K = lambda x: x
# The block below mimicks the defaults in Simon's scripts, and needs to be changed
# when these are updated.
if K is QQ:
cmd = 'ellrank(%s, %s);' % (list(E.ainvs()), [P._pari_() for P in known_points])
if lim1 is None:
lim1 = 5
if lim3 is None:
lim3 = 50
if limtriv is None:
limtriv = 3
else:
cmd = 'bnfellrank(K, %s, %s);' % (list(E.ainvs()), [P._pari_() for P in known_points])
if lim1 is None:
lim1 = 2
if lim3 is None:
lim3 = 4
if limtriv is None:
limtriv = 2
gp('DEBUGLEVEL_ell=%s; LIM1=%s; LIM3=%s; LIMTRIV=%s; MAXPROB=%s; LIMBIGPRIME=%s;'%(
verbose, lim1, lim3, limtriv, maxprob, limbigprime))
if verbose >= 2:
print(cmd)
s = gp.eval('ans=%s;'%cmd)
if s.find(" *** ") != -1:
raise RuntimeError("\n%s\nAn error occurred while running Simon's 2-descent program"%s)
if verbose > 0:
print(s)
v = gp.eval('ans')
if v=='ans': # then the call to ellrank() or bnfellrank() failed
raise RuntimeError("An error occurred while running Simon's 2-descent program")
if verbose >= 2:
print("v = %s" % v)
# pari represents field elements as Mod(poly, defining-poly)
# so this function will return the respective elements of K
def _gp_mod(*args):
return args[0]
ans = sage_eval(v, {'Mod': _gp_mod, 'y': K.gen(0)})
lower = ZZ(ans[0])
upper = ZZ(ans[1])
points = [E_orig([from_K(c) for c in list(P)]) for P in ans[2]]
points = [P for P in points if P.has_infinite_order()]
return lower, upper, points