def __init__(self, degree, base_ring, category=None):
"""
Base class for matrix groups over generic base rings
You should not use this class directly. Instead, use one of
the more specialized derived classes.
INPUT:
- ``degree`` -- integer. The degree (matrix size) of the
matrix group.
- ``base_ring`` -- ring. The base ring of the matrices.
TESTS::
sage: G = GL(2, QQ)
sage: from sage.groups.matrix_gps.matrix_group import MatrixGroup_generic
sage: isinstance(G, MatrixGroup_generic)
True
"""
assert is_Ring(base_ring)
assert is_Integer(degree)
self._deg = degree
if self._deg <= 0:
raise ValueError('the degree must be at least 1')
if (category is None) and is_FiniteField(base_ring):
from sage.categories.finite_groups import FiniteGroups
category = FiniteGroups()
super(MatrixGroup_generic, self).__init__(base=base_ring, category=category)
def create_key(self,
coeff_ring=ZZ,
group='Sp(4,Z)',
weights='even',
degree=2,
default_prec=SMF_DEFAULT_PREC):
"""
Create a key which uniquely defines this algebra of Siegel modular
forms.
TESTS::
sage: SiegelModularFormsAlgebra.create_key(coeff_ring=QQ, group='Gamma0(5)', weights='all', degree=2, default_prec=41)
(SMFAlg{"Gamma0(5)", "all", 2, 41}(FractionField(...)), Integer Ring)
"""
from sage.rings.ring import is_Ring
if not is_Ring(coeff_ring):
raise TypeError('The coefficient ring must be a ring')
if not (isinstance(group, str) or group is None):
raise TypeError('The group must be given by a string, or None')
if not weights in ('even', 'all'):
raise ValueError("The argument weights must be 'even' or 'all'")
try:
degree = int(degree)
except TypeError:
raise TypeError('The degree must be a positive integer')
if degree < 1:
raise ValueError('The degree must be a positive integer')
if degree == 1:
raise ValueError(
'Use ModularForms if you want to work with Siegel modular forms of degree 1'
)
if degree > 2:
raise NotImplementedError(
'Siegel modular forms of degree > 2 are not yet implemented')
try:
default_prec = int(default_prec)
except TypeError:
raise TypeError('The default precision must be a positive integer')
from .pushout import SiegelModularFormsAlgebraFunctor
F = SiegelModularFormsAlgebraFunctor(group=group,
weights=weights,
degree=degree,
default_prec=default_prec)
while hasattr(coeff_ring, 'construction'):
C = coeff_ring.construction()
if C is None:
break
F = F * C[0]
coeff_ring = C[1]
return (F, coeff_ring)
def _coerce_map_from_(self, S):
r"""
True if there is a coercion from ``S`` to ``self``, False otherwise.
The actual coercion is done by the :meth:`_element_constructor_`
method.
INPUT:
- ``S`` - a Sage object.
The objects that coerce into a group algebra `k[G]` are:
- any group algebra `R[H]` as long as `R` coerces into `k` and
`H` coerces into `G`.
- any ring `R` which coerces into `k`
- any group `H` which coerces into either `k` or `G`.
Note that if `H` is a group which coerces into both `k` and
`G`, then Sage will always use the map to `k`. For example,
if `\ZZ` is the ring (or group) of integers, then `\ZZ` will
coerce to any `k[G]`, by sending `\ZZ` to `k`.
EXAMPLES::
sage: A = GroupAlgebra(SymmetricGroup(4), QQ)
sage: B = GroupAlgebra(SymmetricGroup(3), ZZ)
sage: A._coerce_map_from_(B)
True
sage: B._coerce_map_from_(A)
False
sage: A._coerce_map_from_(ZZ)
True
sage: A._coerce_map_from_(CC)
False
sage: A._coerce_map_from_(SymmetricGroup(5))
False
sage: A._coerce_map_from_(SymmetricGroup(2))
True
"""
from sage.rings.ring import is_Ring
from sage.groups.old import Group
k = self.base_ring()
G = self.group()
if isinstance(S, GroupAlgebra):
return (k.has_coerce_map_from(S.base_ring())
and G.has_coerce_map_from(S.group()))
if is_Ring(S):
return k.has_coerce_map_from(S)
if isinstance(S, Group):
return k.has_coerce_map_from(S) or G.has_coerce_map_from(S)
def _coerce_map_from_(self, S):
r"""
True if there is a coercion from ``S`` to ``self``, False otherwise.
The actual coercion is done by the :meth:`_element_constructor_`
method.
INPUT:
- ``S`` - a Sage object.
The objects that coerce into a group algebra `k[G]` are:
- any group algebra `R[H]` as long as `R` coerces into `k` and
`H` coerces into `G`.
- any ring `R` which coerces into `k`
- any group `H` which coerces into either `k` or `G`.
Note that if `H` is a group which coerces into both `k` and
`G`, then Sage will always use the map to `k`. For example,
if `\ZZ` is the ring (or group) of integers, then `\ZZ` will
coerce to any `k[G]`, by sending `\ZZ` to `k`.
EXAMPLES::
sage: A = GroupAlgebra(SymmetricGroup(4), QQ)
sage: B = GroupAlgebra(SymmetricGroup(3), ZZ)
sage: A._coerce_map_from_(B)
True
sage: B._coerce_map_from_(A)
False
sage: A._coerce_map_from_(ZZ)
True
sage: A._coerce_map_from_(CC)
False
sage: A._coerce_map_from_(SymmetricGroup(5))
False
sage: A._coerce_map_from_(SymmetricGroup(2))
True
"""
from sage.rings.ring import is_Ring
from sage.groups.old import Group
k = self.base_ring()
G = self.group()
if isinstance(S, GroupAlgebra):
return (k.has_coerce_map_from(S.base_ring())
and G.has_coerce_map_from(S.group()))
if is_Ring(S):
return k.has_coerce_map_from(S)
if isinstance(S,Group):
return k.has_coerce_map_from(S) or G.has_coerce_map_from(S)
def base_change_to(self, R):
"""
Alters the quadratic form to have all coefficients
defined over the new base_ring R. Here R must be
coercible to from the current base ring.
Note: This is preferable to performing an explicit
coercion through the base_ring() method, which does
not affect the individual coefficients. This is
particularly useful for performing fast modular
arithmetic evaluations.
INPUT:
R -- a ring
OUTPUT:
quadratic form
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ,[1,1]); Q
Quadratic form in 2 variables over Integer Ring with coefficients:
[ 1 0 ]
[ * 1 ]
::
sage: Q1 = Q.base_change_to(IntegerModRing(5)); Q1
Quadratic form in 2 variables over Ring of integers modulo 5 with coefficients:
[ 1 0 ]
[ * 1 ]
sage: Q1([35,11])
1
"""
## Check that a canonical coercion is possible
if not is_Ring(R):
raise TypeError, "Oops! R is not a ring. =("
if not R.has_coerce_map_from(self.base_ring()):
raise TypeError, "Oops! There is no canonical coercion from " + str(
self.base_ring()) + " to R."
## Return the coerced form
return QuadraticForm(R, self.dim(),
[R(x) for x in self.coefficients()])
def base_change_to(self, R):
"""
Alters the quadratic form to have all coefficients
defined over the new base_ring R. Here R must be
coercible to from the current base ring.
Note: This is preferable to performing an explicit
coercion through the base_ring() method, which does
not affect the individual coefficients. This is
particularly useful for performing fast modular
arithmetic evaluations.
INPUT:
R -- a ring
OUTPUT:
quadratic form
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ,[1,1]); Q
Quadratic form in 2 variables over Integer Ring with coefficients:
[ 1 0 ]
[ * 1 ]
::
sage: Q1 = Q.base_change_to(IntegerModRing(5)); Q1
Quadratic form in 2 variables over Ring of integers modulo 5 with coefficients:
[ 1 0 ]
[ * 1 ]
sage: Q1([35,11])
1
"""
## Check that a canonical coercion is possible
if not is_Ring(R):
raise TypeError("Oops! R is not a ring. =(")
if not R.has_coerce_map_from(self.base_ring()):
raise TypeError("Oops! There is no canonical coercion from " + str(self.base_ring()) + " to R.")
## Return the coerced form
return QuadraticForm(R, self.dim(), [R(x) for x in self.coefficients()])
def create_key(self, coeff_ring=ZZ, group='Sp(4,Z)', weights='even', degree=2, default_prec=SMF_DEFAULT_PREC):
"""
Create a key which uniquely defines this algebra of Siegel modular
forms.
TESTS::
sage: SiegelModularFormsAlgebra.create_key(coeff_ring=QQ, group='Gamma0(5)', weights='all', degree=2, default_prec=41)
(SMFAlg{"Gamma0(5)", "all", 2, 41}(FractionField(...)), Integer Ring)
"""
from sage.rings.ring import is_Ring
if not is_Ring(coeff_ring):
raise TypeError, 'The coefficient ring must be a ring'
if not (isinstance(group, str) or group is None):
raise TypeError, 'The group must be given by a string, or None'
if not weights in ('even', 'all'):
raise ValueError, "The argument weights must be 'even' or 'all'"
try:
degree = int(degree)
except TypeError:
raise TypeError, 'The degree must be a positive integer'
if degree < 1:
raise ValueError, 'The degree must be a positive integer'
if degree == 1:
raise ValueError, 'Use ModularForms if you want to work with Siegel modular forms of degree 1'
if degree > 2:
raise NotImplementedError, 'Siegel modular forms of degree > 2 are not yet implemented'
try:
default_prec = int(default_prec)
except TypeError:
raise TypeError, 'The default precision must be a positive integer'
from pushout import SiegelModularFormsAlgebraFunctor
F = SiegelModularFormsAlgebraFunctor(group=group, weights=weights, degree=degree, default_prec=default_prec)
while hasattr(coeff_ring, 'construction'):
C = coeff_ring.construction()
if C is None:
break
F = F * C[0]
coeff_ring = C[1]
return (F, coeff_ring)
def LaurentPolynomialRing(base_ring,
arg1=None,
arg2=None,
sparse=False,
order='degrevlex',
names=None,
name=None):
r"""
Return the globally unique univariate or multivariate Laurent polynomial
ring with given properties and variable name or names.
There are four ways to call the Laurent polynomial ring constructor:
1. ``LaurentPolynomialRing(base_ring, name, sparse=False)``
2. ``LaurentPolynomialRing(base_ring, names, order='degrevlex')``
3. ``LaurentPolynomialRing(base_ring, name, n, order='degrevlex')``
4. ``LaurentPolynomialRing(base_ring, n, name, order='degrevlex')``
The optional arguments sparse and order *must* be explicitly
named, and the other arguments must be given positionally.
INPUT:
- ``base_ring`` -- a commutative ring
- ``name`` -- a string
- ``names`` -- a list or tuple of names, or a comma separated string
- ``n`` -- a positive integer
- ``sparse`` -- bool (default: False), whether or not elements are sparse
- ``order`` -- string or
:class:`~sage.rings.polynomial.term_order.TermOrder`, e.g.,
- ``'degrevlex'`` (default) -- degree reverse lexicographic
- ``'lex'`` -- lexicographic
- ``'deglex'`` -- degree lexicographic
- ``TermOrder('deglex',3) + TermOrder('deglex',3)`` -- block ordering
OUTPUT:
``LaurentPolynomialRing(base_ring, name, sparse=False)`` returns a
univariate Laurent polynomial ring; all other input formats return a
multivariate Laurent polynomial ring.
UNIQUENESS and IMMUTABILITY: In Sage there is exactly one
single-variate Laurent polynomial ring over each base ring in each choice
of variable and sparseness. There is also exactly one multivariate
Laurent polynomial ring over each base ring for each choice of names of
variables and term order.
::
sage: R.<x,y> = LaurentPolynomialRing(QQ,2); R
Multivariate Laurent Polynomial Ring in x, y over Rational Field
sage: f = x^2 - 2*y^-2
You can't just globally change the names of those variables.
This is because objects all over Sage could have pointers to
that polynomial ring.
::
sage: R._assign_names(['z','w'])
Traceback (most recent call last):
...
ValueError: variable names cannot be changed after object creation.
EXAMPLES:
1. ``LaurentPolynomialRing(base_ring, name, sparse=False)``
::
sage: LaurentPolynomialRing(QQ, 'w')
Univariate Laurent Polynomial Ring in w over Rational Field
Use the diamond brackets notation to make the variable
ready for use after you define the ring::
sage: R.<w> = LaurentPolynomialRing(QQ)
sage: (1 + w)^3
1 + 3*w + 3*w^2 + w^3
You must specify a name::
sage: LaurentPolynomialRing(QQ)
Traceback (most recent call last):
...
TypeError: You must specify the names of the variables.
sage: R.<abc> = LaurentPolynomialRing(QQ, sparse=True); R
Univariate Laurent Polynomial Ring in abc over Rational Field
sage: R.<w> = LaurentPolynomialRing(PolynomialRing(GF(7),'k')); R
Univariate Laurent Polynomial Ring in w over Univariate Polynomial Ring in k over Finite Field of size 7
Rings with different variables are different::
sage: LaurentPolynomialRing(QQ, 'x') == LaurentPolynomialRing(QQ, 'y')
False
2. ``LaurentPolynomialRing(base_ring, names, order='degrevlex')``
::
sage: R = LaurentPolynomialRing(QQ, 'a,b,c'); R
Multivariate Laurent Polynomial Ring in a, b, c over Rational Field
sage: S = LaurentPolynomialRing(QQ, ['a','b','c']); S
Multivariate Laurent Polynomial Ring in a, b, c over Rational Field
sage: T = LaurentPolynomialRing(QQ, ('a','b','c')); T
Multivariate Laurent Polynomial Ring in a, b, c over Rational Field
All three rings are identical.
::
sage: (R is S) and (S is T)
True
There is a unique Laurent polynomial ring with each term order::
sage: R = LaurentPolynomialRing(QQ, 'x,y,z', order='degrevlex'); R
Multivariate Laurent Polynomial Ring in x, y, z over Rational Field
sage: S = LaurentPolynomialRing(QQ, 'x,y,z', order='invlex'); S
Multivariate Laurent Polynomial Ring in x, y, z over Rational Field
sage: S is LaurentPolynomialRing(QQ, 'x,y,z', order='invlex')
True
sage: R == S
False
3. ``LaurentPolynomialRing(base_ring, name, n, order='degrevlex')``
If you specify a single name as a string and a number of
variables, then variables labeled with numbers are created.
::
sage: LaurentPolynomialRing(QQ, 'x', 10)
Multivariate Laurent Polynomial Ring in x0, x1, x2, x3, x4, x5, x6, x7, x8, x9 over Rational Field
sage: LaurentPolynomialRing(GF(7), 'y', 5)
Multivariate Laurent Polynomial Ring in y0, y1, y2, y3, y4 over Finite Field of size 7
sage: LaurentPolynomialRing(QQ, 'y', 3, sparse=True)
Multivariate Laurent Polynomial Ring in y0, y1, y2 over Rational Field
By calling the
:meth:`~sage.structure.category_object.CategoryObject.inject_variables`
method, all those variable names are available for interactive use::
sage: R = LaurentPolynomialRing(GF(7),15,'w'); R
Multivariate Laurent Polynomial Ring in w0, w1, w2, w3, w4, w5, w6, w7, w8, w9, w10, w11, w12, w13, w14 over Finite Field of size 7
sage: R.inject_variables()
Defining w0, w1, w2, w3, w4, w5, w6, w7, w8, w9, w10, w11, w12, w13, w14
sage: (w0 + 2*w8 + w13)^2
w0^2 + 4*w0*w8 + 4*w8^2 + 2*w0*w13 + 4*w8*w13 + w13^2
"""
if is_Element(arg1) and not isinstance(arg1, integer_types + (Integer, )):
arg1 = repr(arg1)
if is_Element(arg2) and not isinstance(arg2, integer_types + (Integer, )):
arg2 = repr(arg2)
if isinstance(arg1, integer_types + (Integer, )):
arg1, arg2 = arg2, arg1
if not names is None:
arg1 = names
elif not name is None:
arg1 = name
if not is_Ring(base_ring):
raise TypeError('base_ring must be a ring')
if arg1 is None:
raise TypeError("You must specify the names of the variables.")
R = None
if isinstance(arg1, (list, tuple)):
arg1 = [str(x) for x in arg1]
if isinstance(arg2, (list, tuple)):
arg2 = [str(x) for x in arg2]
if isinstance(arg2, integer_types + (Integer, )):
# 3. LaurentPolynomialRing(base_ring, names, n, order='degrevlex'):
if not isinstance(arg1, (list, tuple, str)):
raise TypeError("You *must* specify the names of the variables.")
n = int(arg2)
names = arg1
R = _multi_variate(base_ring, names, n, sparse, order)
elif isinstance(arg1,
str) or (isinstance(arg1, (list, tuple))
and len(arg1) == 1) and isinstance(arg1[0], str):
if isinstance(arg1, (list, tuple)):
arg1 = arg1[0]
if not ',' in arg1:
# 1. LaurentPolynomialRing(base_ring, name, sparse=False):
if not arg2 is None:
raise TypeError(
"if second arguments is a string with no commas, then there must be no other non-optional arguments"
)
name = arg1
R = _single_variate(base_ring, name, sparse)
else:
# 2-4. LaurentPolynomialRing(base_ring, names, order='degrevlex'):
if not arg2 is None:
raise TypeError(
"invalid input to LaurentPolynomialRing function; please see the docstring for that function"
)
names = arg1.split(',')
n = len(names)
R = _multi_variate(base_ring, names, n, sparse, order)
elif isinstance(arg1, (list, tuple)):
# LaurentPolynomialRing(base_ring, names (list or tuple), order='degrevlex'):
names = arg1
n = len(names)
R = _multi_variate(base_ring, names, n, sparse, order)
if arg1 is None and arg2 is None:
raise TypeError("you *must* specify the indeterminates (as not None).")
if R is None:
raise TypeError(
"invalid input (%s, %s, %s) to PolynomialRing function; please see the docstring for that function"
% (base_ring, arg1, arg2))
return R
def _element_constructor_(self, x):
r"""
Try to turn ``x`` into an element of ``self``.
INPUT:
- ``x`` - an element of some group algebra or of a
ring or of a group
OUTPUT: ``x`` as a member of ``self``.
sage: G = KleinFourGroup()
sage: f = G.gen(0)
sage: ZG = GroupAlgebra(G)
sage: ZG(f) # indirect doctest
(3,4)
sage: ZG(1) == ZG(G(1))
True
sage: G = AbelianGroup(1)
sage: ZG = GroupAlgebra(G)
sage: f = ZG.group().gen()
sage: ZG(FormalSum([(1,f), (2, f**2)]))
f + 2*f^2
sage: G = GL(2,7)
sage: OG = GroupAlgebra(G, ZZ[sqrt(5)])
sage: OG(2)
2*[1 0]
[0 1]
sage: OG(G(2)) # conversion is not the obvious one
[2 0]
[0 2]
sage: OG(FormalSum([ (1, G(2)), (2, RR(0.77)) ]) )
Traceback (most recent call last):
...
TypeError: Attempt to coerce non-integral RealNumber to Integer
sage: OG(OG.base_ring().basis()[1])
sqrt5*[1 0]
[0 1]
"""
from sage.rings.ring import is_Ring
from sage.groups.group import is_Group
from sage.structure.formal_sum import FormalSum
k = self.base_ring()
G = self.group()
S = x.parent()
if isinstance(S, GroupAlgebra):
if self.has_coerce_map_from(S):
# coerce monomials, coerce coefficients, reassemble
d = x.monomial_coefficients()
new_d = {}
for g in d:
g1 = G(g)
if g1 in new_d:
new_d[g1] += k(d[g]) + new_d[g1]
else:
new_d[g1] = k(d[g])
return self._from_dict(new_d)
elif is_Ring(S):
# coerce to multiple of identity element
return k(x) * self(1)
elif is_Group(S):
# Check whether group coerces to base_ring first.
if k.has_coerce_map_from(S):
return k(x) * self(1)
if G.has_coerce_map_from(S):
return self.monomial(self.group()(x))
elif isinstance(x, FormalSum) and k.has_coerce_map_from(S.base_ring()):
y = [(G(g), k(coeff)) for coeff, g in x]
return self.sum_of_terms(y)
raise TypeError("Don't know how to create an element of %s from %s" % \
(self, x))
def PolynomialRing(base_ring, *args, **kwds):
r"""
Return the globally unique univariate or multivariate polynomial
ring with given properties and variable name or names.
There are many ways to specify the variables for the polynomial ring:
1. ``PolynomialRing(base_ring, name, ...)``
2. ``PolynomialRing(base_ring, names, ...)``
3. ``PolynomialRing(base_ring, n, names, ...)``
4. ``PolynomialRing(base_ring, n, ..., var_array=var_array, ...)``
The ``...`` at the end of these commands stands for additional
keywords, like ``sparse`` or ``order``.
INPUT:
- ``base_ring`` -- a ring
- ``n`` -- an integer
- ``name`` -- a string
- ``names`` -- a list or tuple of names (strings), or a comma separated string
- ``var_array`` -- a list or tuple of names, or a comma separated string
- ``sparse`` -- bool: whether or not elements are sparse. The
default is a dense representation (``sparse=False``) for
univariate rings and a sparse representation (``sparse=True``)
for multivariate rings.
- ``order`` -- string or
:class:`~sage.rings.polynomial.term_order.TermOrder` object, e.g.,
- ``'degrevlex'`` (default) -- degree reverse lexicographic
- ``'lex'`` -- lexicographic
- ``'deglex'`` -- degree lexicographic
- ``TermOrder('deglex',3) + TermOrder('deglex',3)`` -- block ordering
- ``implementation`` -- string or None; selects an implementation in cases
where Sage includes multiple choices (currently `\ZZ[x]` can be
implemented with ``'NTL'`` or ``'FLINT'``; default is ``'FLINT'``).
For many base rings, the ``"singular"`` implementation is available.
One can always specify ``implementation="generic"`` for a generic
Sage implementation which does not use any specialized library.
.. NOTE::
If the given implementation does not exist for rings with the given
number of generators and the given sparsity, then an error results.
OUTPUT:
``PolynomialRing(base_ring, name, sparse=False)`` returns a univariate
polynomial ring; also, PolynomialRing(base_ring, names, sparse=False)
yields a univariate polynomial ring, if names is a list or tuple
providing exactly one name. All other input formats return a
multivariate polynomial ring.
UNIQUENESS and IMMUTABILITY: In Sage there is exactly one
single-variate polynomial ring over each base ring in each choice
of variable, sparseness, and implementation. There is also exactly
one multivariate polynomial ring over each base ring for each
choice of names of variables and term order. The names of the
generators can only be temporarily changed after the ring has been
created. Do this using the localvars context:
EXAMPLES:
**1. PolynomialRing(base_ring, name, ...)**
::
sage: PolynomialRing(QQ, 'w')
Univariate Polynomial Ring in w over Rational Field
sage: PolynomialRing(QQ, name='w')
Univariate Polynomial Ring in w over Rational Field
Use the diamond brackets notation to make the variable
ready for use after you define the ring::
sage: R.<w> = PolynomialRing(QQ)
sage: (1 + w)^3
w^3 + 3*w^2 + 3*w + 1
You must specify a name::
sage: PolynomialRing(QQ)
Traceback (most recent call last):
...
TypeError: you must specify the names of the variables
sage: R.<abc> = PolynomialRing(QQ, sparse=True); R
Sparse Univariate Polynomial Ring in abc over Rational Field
sage: R.<w> = PolynomialRing(PolynomialRing(GF(7),'k')); R
Univariate Polynomial Ring in w over Univariate Polynomial Ring in k over Finite Field of size 7
The square bracket notation::
sage: R.<y> = QQ['y']; R
Univariate Polynomial Ring in y over Rational Field
sage: y^2 + y
y^2 + y
In fact, since the diamond brackets on the left determine the
variable name, you can omit the variable from the square brackets::
sage: R.<zz> = QQ[]; R
Univariate Polynomial Ring in zz over Rational Field
sage: (zz + 1)^2
zz^2 + 2*zz + 1
This is exactly the same ring as what PolynomialRing returns::
sage: R is PolynomialRing(QQ,'zz')
True
However, rings with different variables are different::
sage: QQ['x'] == QQ['y']
False
Sage has two implementations of univariate polynomials over the
integers, one based on NTL and one based on FLINT. The default
is FLINT. Note that FLINT uses a "more dense" representation for
its polynomials than NTL, so in particular, creating a polynomial
like 2^1000000 * x^1000000 in FLINT may be unwise.
::
sage: ZxNTL = PolynomialRing(ZZ, 'x', implementation='NTL'); ZxNTL
Univariate Polynomial Ring in x over Integer Ring (using NTL)
sage: ZxFLINT = PolynomialRing(ZZ, 'x', implementation='FLINT'); ZxFLINT
Univariate Polynomial Ring in x over Integer Ring
sage: ZxFLINT is ZZ['x']
True
sage: ZxFLINT is PolynomialRing(ZZ, 'x')
True
sage: xNTL = ZxNTL.gen()
sage: xFLINT = ZxFLINT.gen()
sage: xNTL.parent()
Univariate Polynomial Ring in x over Integer Ring (using NTL)
sage: xFLINT.parent()
Univariate Polynomial Ring in x over Integer Ring
There is a coercion from the non-default to the default
implementation, so the values can be mixed in a single
expression::
sage: (xNTL + xFLINT^2)
x^2 + x
The result of such an expression will use the default, i.e.,
the FLINT implementation::
sage: (xNTL + xFLINT^2).parent()
Univariate Polynomial Ring in x over Integer Ring
The generic implementation uses neither NTL nor FLINT::
sage: Zx = PolynomialRing(ZZ, 'x', implementation='generic'); Zx
Univariate Polynomial Ring in x over Integer Ring
sage: Zx.element_class
<... 'sage.rings.polynomial.polynomial_element.Polynomial_generic_dense'>
**2. PolynomialRing(base_ring, names, ...)**
::
sage: R = PolynomialRing(QQ, 'a,b,c'); R
Multivariate Polynomial Ring in a, b, c over Rational Field
sage: S = PolynomialRing(QQ, ['a','b','c']); S
Multivariate Polynomial Ring in a, b, c over Rational Field
sage: T = PolynomialRing(QQ, ('a','b','c')); T
Multivariate Polynomial Ring in a, b, c over Rational Field
All three rings are identical::
sage: R is S
True
sage: S is T
True
There is a unique polynomial ring with each term order::
sage: R = PolynomialRing(QQ, 'x,y,z', order='degrevlex'); R
Multivariate Polynomial Ring in x, y, z over Rational Field
sage: S = PolynomialRing(QQ, 'x,y,z', order='invlex'); S
Multivariate Polynomial Ring in x, y, z over Rational Field
sage: S is PolynomialRing(QQ, 'x,y,z', order='invlex')
True
sage: R == S
False
Note that a univariate polynomial ring is returned, if the list
of names is of length one. If it is of length zero, a multivariate
polynomial ring with no variables is returned.
::
sage: PolynomialRing(QQ,["x"])
Univariate Polynomial Ring in x over Rational Field
sage: PolynomialRing(QQ,[])
Multivariate Polynomial Ring in no variables over Rational Field
The Singular implementation always returns a multivariate ring,
even for 1 variable::
sage: PolynomialRing(QQ, "x", implementation="singular")
Multivariate Polynomial Ring in x over Rational Field
sage: P.<x> = PolynomialRing(QQ, implementation="singular"); P
Multivariate Polynomial Ring in x over Rational Field
**3. PolynomialRing(base_ring, n, names, ...)** (where the arguments
``n`` and ``names`` may be reversed)
If you specify a single name as a string and a number of
variables, then variables labeled with numbers are created.
::
sage: PolynomialRing(QQ, 'x', 10)
Multivariate Polynomial Ring in x0, x1, x2, x3, x4, x5, x6, x7, x8, x9 over Rational Field
sage: PolynomialRing(QQ, 2, 'alpha0')
Multivariate Polynomial Ring in alpha00, alpha01 over Rational Field
sage: PolynomialRing(GF(7), 'y', 5)
Multivariate Polynomial Ring in y0, y1, y2, y3, y4 over Finite Field of size 7
sage: PolynomialRing(QQ, 'y', 3, sparse=True)
Multivariate Polynomial Ring in y0, y1, y2 over Rational Field
Note that a multivariate polynomial ring is returned when an
explicit number is given.
::
sage: PolynomialRing(QQ,"x",1)
Multivariate Polynomial Ring in x over Rational Field
sage: PolynomialRing(QQ,"x",0)
Multivariate Polynomial Ring in no variables over Rational Field
It is easy in Python to create fairly arbitrary variable names. For
example, here is a ring with generators labeled by the primes less
than 100::
sage: R = PolynomialRing(ZZ, ['x%s'%p for p in primes(100)]); R
Multivariate Polynomial Ring in x2, x3, x5, x7, x11, x13, x17, x19, x23, x29, x31, x37, x41, x43, x47, x53, x59, x61, x67, x71, x73, x79, x83, x89, x97 over Integer Ring
By calling the
:meth:`~sage.structure.category_object.CategoryObject.inject_variables`
method, all those variable names are available for interactive use::
sage: R.inject_variables()
Defining x2, x3, x5, x7, x11, x13, x17, x19, x23, x29, x31, x37, x41, x43, x47, x53, x59, x61, x67, x71, x73, x79, x83, x89, x97
sage: (x2 + x41 + x71)^2
x2^2 + 2*x2*x41 + x41^2 + 2*x2*x71 + 2*x41*x71 + x71^2
**4. PolynomialRing(base_ring, n, ..., var_array=var_array, ...)**
This creates an array of variables where each variables begins with an
entry in ``var_array`` and is indexed from 0 to `n-1`. ::
sage: PolynomialRing(ZZ, 3, var_array=['x','y'])
Multivariate Polynomial Ring in x0, y0, x1, y1, x2, y2 over Integer Ring
sage: PolynomialRing(ZZ, 3, var_array='a,b')
Multivariate Polynomial Ring in a0, b0, a1, b1, a2, b2 over Integer Ring
It is possible to create higher-dimensional arrays::
sage: PolynomialRing(ZZ, 2, 3, var_array=('p', 'q'))
Multivariate Polynomial Ring in p00, q00, p01, q01, p02, q02, p10, q10, p11, q11, p12, q12 over Integer Ring
sage: PolynomialRing(ZZ, 2, 3, 4, var_array='m')
Multivariate Polynomial Ring in m000, m001, m002, m003, m010, m011, m012, m013, m020, m021, m022, m023, m100, m101, m102, m103, m110, m111, m112, m113, m120, m121, m122, m123 over Integer Ring
The array is always at least 2-dimensional. So, if
``var_array`` is a single string and only a single number `n`
is given, this creates an `n \times n` array of variables::
sage: PolynomialRing(ZZ, 2, var_array='m')
Multivariate Polynomial Ring in m00, m01, m10, m11 over Integer Ring
**Square brackets notation**
You can alternatively create a polynomial ring over a ring `R` with
square brackets::
sage: RR["x"]
Univariate Polynomial Ring in x over Real Field with 53 bits of precision
sage: RR["x,y"]
Multivariate Polynomial Ring in x, y over Real Field with 53 bits of precision
sage: P.<x,y> = RR[]; P
Multivariate Polynomial Ring in x, y over Real Field with 53 bits of precision
This notation does not allow to set any of the optional arguments.
**Changing variable names**
Consider ::
sage: R.<x,y> = PolynomialRing(QQ,2); R
Multivariate Polynomial Ring in x, y over Rational Field
sage: f = x^2 - 2*y^2
You can't just globally change the names of those variables.
This is because objects all over Sage could have pointers to
that polynomial ring. ::
sage: R._assign_names(['z','w'])
Traceback (most recent call last):
...
ValueError: variable names cannot be changed after object creation.
However, you can very easily change the names within a ``with`` block::
sage: with localvars(R, ['z','w']):
....: print(f)
z^2 - 2*w^2
After the ``with`` block the names revert to what they were before::
sage: print(f)
x^2 - 2*y^2
TESTS:
We test here some changes introduced in :trac:`9944`.
If there is no dense implementation for the given number of
variables, then requesting a dense ring is an error::
sage: S.<x,y> = PolynomialRing(QQ, sparse=False)
Traceback (most recent call last):
...
NotImplementedError: a dense representation of multivariate polynomials is not supported
Check uniqueness if the same implementation is used for different
values of the ``"implementation"`` keyword::
sage: R = PolynomialRing(QQbar, 'j', implementation="generic")
sage: S = PolynomialRing(QQbar, 'j', implementation=None)
sage: R is S
True
sage: R = PolynomialRing(ZZ['t'], 'j', implementation="generic")
sage: S = PolynomialRing(ZZ['t'], 'j', implementation=None)
sage: R is S
True
sage: R = PolynomialRing(QQbar, 'j,k', implementation="generic")
sage: S = PolynomialRing(QQbar, 'j,k', implementation=None)
sage: R is S
True
sage: R = PolynomialRing(ZZ, 'j,k', implementation="singular")
sage: S = PolynomialRing(ZZ, 'j,k', implementation=None)
sage: R is S
True
sage: R = PolynomialRing(ZZ, 'p', sparse=True, implementation="generic")
sage: S = PolynomialRing(ZZ, 'p', sparse=True)
sage: R is S
True
The generic implementation is different in some cases::
sage: R = PolynomialRing(GF(2), 'j', implementation="generic"); type(R)
<class 'sage.rings.polynomial.polynomial_ring.PolynomialRing_field_with_category'>
sage: S = PolynomialRing(GF(2), 'j'); type(S)
<class 'sage.rings.polynomial.polynomial_ring.PolynomialRing_dense_mod_p_with_category'>
sage: R = PolynomialRing(ZZ, 'x,y', implementation="generic"); type(R)
<class 'sage.rings.polynomial.multi_polynomial_ring.MPolynomialRing_polydict_domain_with_category'>
sage: S = PolynomialRing(ZZ, 'x,y'); type(S)
<type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomialRing_libsingular'>
Sparse univariate polynomials only support a generic
implementation::
sage: R = PolynomialRing(ZZ, 'j', sparse=True); type(R)
<class 'sage.rings.polynomial.polynomial_ring.PolynomialRing_integral_domain_with_category'>
sage: R = PolynomialRing(GF(49), 'j', sparse=True); type(R)
<class 'sage.rings.polynomial.polynomial_ring.PolynomialRing_field_with_category'>
If the requested implementation is not known or not supported for
the given arguments, then an error results::
sage: R.<x0> = PolynomialRing(ZZ, implementation='Foo')
Traceback (most recent call last):
...
ValueError: unknown implementation 'Foo' for dense polynomial rings over Integer Ring
sage: R.<x0> = PolynomialRing(GF(2), implementation='GF2X', sparse=True)
Traceback (most recent call last):
...
ValueError: unknown implementation 'GF2X' for sparse polynomial rings over Finite Field of size 2
sage: R.<x,y> = PolynomialRing(ZZ, implementation='FLINT')
Traceback (most recent call last):
...
ValueError: unknown implementation 'FLINT' for multivariate polynomial rings
sage: R.<x> = PolynomialRing(QQbar, implementation="whatever")
Traceback (most recent call last):
...
ValueError: unknown implementation 'whatever' for dense polynomial rings over Algebraic Field
sage: R.<x> = PolynomialRing(ZZ['t'], implementation="whatever")
Traceback (most recent call last):
...
ValueError: unknown implementation 'whatever' for dense polynomial rings over Univariate Polynomial Ring in t over Integer Ring
sage: PolynomialRing(RR, "x,y", implementation="whatever")
Traceback (most recent call last):
...
ValueError: unknown implementation 'whatever' for multivariate polynomial rings
sage: PolynomialRing(RR, name="x", implementation="singular")
Traceback (most recent call last):
...
NotImplementedError: polynomials over Real Field with 53 bits of precision are not supported in Singular
The following corner case used to result in a warning message from
``libSingular``, and the generators of the resulting polynomial
ring were not zero::
sage: R = Integers(1)['x','y']
sage: R.0 == 0
True
We verify that :trac:`13187` is fixed::
sage: var('t')
t
sage: PolynomialRing(ZZ, name=t) == PolynomialRing(ZZ, name='t')
True
We verify that polynomials with interval coefficients from
:trac:`7712` and :trac:`13760` are fixed::
sage: P.<y,z> = PolynomialRing(RealIntervalField(2))
sage: Q.<x> = PolynomialRing(P)
sage: C = (y-x)^3
sage: C(y/2)
1.?*y^3
sage: R.<x,y> = PolynomialRing(RIF,2)
sage: RIF(-2,1)*x
0.?e1*x
For historical reasons, we allow redundant variable names with the
angle bracket notation. The names must be consistent though! ::
sage: P.<x,y> = PolynomialRing(ZZ, "x,y"); P
Multivariate Polynomial Ring in x, y over Integer Ring
sage: P.<x,y> = ZZ["x,y"]; P
Multivariate Polynomial Ring in x, y over Integer Ring
sage: P.<x,y> = PolynomialRing(ZZ, 2, "x"); P
Traceback (most recent call last):
...
TypeError: variable names specified twice inconsistently: ('x0', 'x1') and ('x', 'y')
We test a lot of invalid input::
sage: PolynomialRing(4)
Traceback (most recent call last):
...
TypeError: base_ring 4 must be a ring
sage: PolynomialRing(QQ, -1)
Traceback (most recent call last):
...
ValueError: number of variables must be non-negative
sage: PolynomialRing(QQ, 1)
Traceback (most recent call last):
...
TypeError: you must specify the names of the variables
sage: PolynomialRing(QQ, "x", None)
Traceback (most recent call last):
...
TypeError: invalid arguments ('x', None) for PolynomialRing
sage: PolynomialRing(QQ, "x", "y")
Traceback (most recent call last):
...
TypeError: variable names specified twice: 'x' and 'y'
sage: PolynomialRing(QQ, 1, "x", 2)
Traceback (most recent call last):
...
TypeError: number of variables specified twice: 1 and 2
sage: PolynomialRing(QQ, "x", names="x")
Traceback (most recent call last):
...
TypeError: variable names specified twice inconsistently: ('x',) and 'x'
sage: PolynomialRing(QQ, name="x", names="x")
Traceback (most recent call last):
...
TypeError: keyword argument 'name' cannot be combined with 'names'
sage: PolynomialRing(QQ, var_array='x')
Traceback (most recent call last):
...
TypeError: you must specify the number of the variables
sage: PolynomialRing(QQ, 2, 'x', var_array='x')
Traceback (most recent call last):
...
TypeError: unable to convert 'x' to an integer
"""
if not ring.is_Ring(base_ring):
raise TypeError("base_ring {!r} must be a ring".format(base_ring))
n = -1 # Unknown number of variables
names = None # Unknown variable names
# Use a single-variate ring by default unless the "singular"
# implementation is asked.
multivariate = kwds.get("implementation") == "singular"
# Check specifically for None because it is an easy mistake to
# make and Integer(None) returns 0, so we wouldn't catch this
# otherwise.
if any(arg is None for arg in args):
raise TypeError(
"invalid arguments {!r} for PolynomialRing".format(args))
if "var_array" in kwds:
for forbidden in "name", "names":
if forbidden in kwds:
raise TypeError(
"keyword argument '%s' cannot be combined with 'var_array'"
% forbidden)
names = kwds.pop("var_array")
if isinstance(names, (tuple, list)):
# Input is a 1-dimensional array
dim = 1
else:
# Input is a 0-dimensional (if a single string was given)
# or a 1-dimensional array
names = normalize_names(-1, names)
dim = len(names) > 1
multivariate = True
if not args:
raise TypeError("you must specify the number of the variables")
# The total dimension must be at least 2
if len(args) == 1 and not dim:
args = [args[0], args[0]]
# All arguments in *args should be a number of variables
suffixes = [""]
for arg in args:
k = Integer(arg)
if k < 0:
raise ValueError("number of variables must be non-negative")
suffixes = [s + str(i) for s in suffixes for i in range(k)]
names = [v + s for s in suffixes for v in names]
else: # No "var_array" keyword
if "name" in kwds:
if "names" in kwds:
raise TypeError(
"keyword argument 'name' cannot be combined with 'names'")
names = [kwds.pop("name")]
# Interpret remaining arguments in *args as either a number of
# variables or as variable names
for arg in args:
try:
k = Integer(arg)
except TypeError:
# Interpret arg as names
if names is not None:
raise TypeError(
"variable names specified twice: %r and %r" %
(names, arg))
names = arg
else:
# Interpret arg as number of variables
if n >= 0:
raise TypeError(
"number of variables specified twice: %r and %r" %
(n, arg))
if k < 0:
raise ValueError(
"number of variables must be non-negative")
n = k
# If number of variables was explicitly given, always
# return a multivariate ring
multivariate = True
if names is None:
try:
names = kwds.pop("names")
except KeyError:
raise TypeError("you must specify the names of the variables")
names = normalize_names(n, names)
# At this point, we have only handled the "names" keyword if it was
# needed. Since we know the variable names, it would logically be
# an error to specify an additional "names" keyword. However,
# people often abuse the preparser with
# R.<x> = PolynomialRing(QQ, 'x')
# and we allow this for historical reasons. However, the names
# must be consistent!
if "names" in kwds:
kwnames = kwds.pop("names")
if kwnames != names:
raise TypeError(
"variable names specified twice inconsistently: %r and %r" %
(names, kwnames))
if multivariate or len(names) != 1:
return _multi_variate(base_ring, names, **kwds)
else:
return _single_variate(base_ring, names, **kwds)
def EllipticCurve(x=None, y=None, j=None, minimal_twist=True):
r"""
Construct an elliptic curve.
In Sage, an elliptic curve is always specified by its a-invariants
.. math::
y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.
INPUT:
There are several ways to construct an elliptic curve:
- ``EllipticCurve([a1,a2,a3,a4,a6])``: Elliptic curve with given
a-invariants. The invariants are coerced into the parent of the
first element. If all are integers, they are coerced into the
rational numbers.
- ``EllipticCurve([a4,a6])``: Same as above, but `a_1=a_2=a_3=0`.
- ``EllipticCurve(label)``: Returns the elliptic curve over Q from
the Cremona database with the given label. The label is a
string, such as ``"11a"`` or ``"37b2"``. The letters in the
label *must* be lower case (Cremona's new labeling).
- ``EllipticCurve(R, [a1,a2,a3,a4,a6])``: Create the elliptic
curve over ``R`` with given a-invariants. Here ``R`` can be an
arbitrary ring. Note that addition need not be defined.
- ``EllipticCurve(j=j0)`` or ``EllipticCurve_from_j(j0)``: Return
an elliptic curve with j-invariant ``j0``.
- ``EllipticCurve(polynomial)``: Read off the a-invariants from
the polynomial coefficients, see
:func:`EllipticCurve_from_Weierstrass_polynomial`.
In each case above where the input is a list of length 2 or 5, one
can instead give a 2 or 5-tuple instead.
EXAMPLES:
We illustrate creating elliptic curves::
sage: EllipticCurve([0,0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
We create a curve from a Cremona label::
sage: EllipticCurve('37b2')
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 1873*x - 31833 over Rational Field
sage: EllipticCurve('5077a')
Elliptic Curve defined by y^2 + y = x^3 - 7*x + 6 over Rational Field
sage: EllipticCurve('389a')
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
Old Cremona labels are allowed::
sage: EllipticCurve('2400FF')
Elliptic Curve defined by y^2 = x^3 + x^2 + 2*x + 8 over Rational Field
Unicode labels are allowed::
sage: EllipticCurve(u'389a')
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
We create curves over a finite field as follows::
sage: EllipticCurve([GF(5)(0),0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5
sage: EllipticCurve(GF(5), [0, 0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5
Elliptic curves over `\ZZ/N\ZZ` with `N` prime are of type
"elliptic curve over a finite field"::
sage: F = Zmod(101)
sage: EllipticCurve(F, [2, 3])
Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Ring of integers modulo 101
sage: E = EllipticCurve([F(2), F(3)])
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_finite_field.EllipticCurve_finite_field_with_category'>
sage: E.category()
Category of schemes over Ring of integers modulo 101
In contrast, elliptic curves over `\ZZ/N\ZZ` with `N` composite
are of type "generic elliptic curve"::
sage: F = Zmod(95)
sage: EllipticCurve(F, [2, 3])
Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Ring of integers modulo 95
sage: E = EllipticCurve([F(2), F(3)])
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_generic.EllipticCurve_generic_with_category'>
sage: E.category()
Category of schemes over Ring of integers modulo 95
The following is a curve over the complex numbers::
sage: E = EllipticCurve(CC, [0,0,1,-1,0])
sage: E
Elliptic Curve defined by y^2 + 1.00000000000000*y = x^3 + (-1.00000000000000)*x over Complex Field with 53 bits of precision
sage: E.j_invariant()
2988.97297297297
We can also create elliptic curves by giving the Weierstrass equation::
sage: x, y = var('x,y')
sage: EllipticCurve(y^2 + y == x^3 + x - 9)
Elliptic Curve defined by y^2 + y = x^3 + x - 9 over Rational Field
sage: R.<x,y> = GF(5)[]
sage: EllipticCurve(x^3 + x^2 + 2 - y^2 - y*x)
Elliptic Curve defined by y^2 + x*y = x^3 + x^2 + 2 over Finite Field of size 5
We can explicitly specify the `j`-invariant::
sage: E = EllipticCurve(j=1728); E; E.j_invariant(); E.label()
Elliptic Curve defined by y^2 = x^3 - x over Rational Field
1728
'32a2'
sage: E = EllipticCurve(j=GF(5)(2)); E; E.j_invariant()
Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 5
2
See :trac:`6657` ::
sage: EllipticCurve(GF(144169),j=1728)
Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 144169
By default, when a rational value of `j` is given, the constructed
curve is a minimal twist (minimal conductor for curves with that
`j`-invariant). This can be changed by setting the optional
parameter ``minimal_twist``, which is True by default, to False::
sage: EllipticCurve(j=100)
Elliptic Curve defined by y^2 = x^3 + x^2 + 3392*x + 307888 over Rational Field
sage: E =EllipticCurve(j=100); E
Elliptic Curve defined by y^2 = x^3 + x^2 + 3392*x + 307888 over Rational Field
sage: E.conductor()
33129800
sage: E.j_invariant()
100
sage: E =EllipticCurve(j=100, minimal_twist=False); E
Elliptic Curve defined by y^2 = x^3 + 488400*x - 530076800 over Rational Field
sage: E.conductor()
298168200
sage: E.j_invariant()
100
Without this option, constructing the curve could take a long time
since both `j` and `j-1728` have to be factored to compute the
minimal twist (see :trac:`13100`)::
sage: E = EllipticCurve_from_j(2^256+1,minimal_twist=False)
sage: E.j_invariant() == 2^256+1
True
TESTS::
sage: R = ZZ['u', 'v']
sage: EllipticCurve(R, [1,1])
Elliptic Curve defined by y^2 = x^3 + x + 1 over Multivariate Polynomial Ring in u, v
over Integer Ring
We create a curve and a point over QQbar (see #6879)::
sage: E = EllipticCurve(QQbar,[0,1])
sage: E(0)
(0 : 1 : 0)
sage: E.base_field()
Algebraic Field
sage: E = EllipticCurve(RR,[1,2]); E; E.base_field()
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 2.00000000000000 over Real Field with 53 bits of precision
Real Field with 53 bits of precision
sage: EllipticCurve(CC,[3,4]); E; E.base_field()
Elliptic Curve defined by y^2 = x^3 + 3.00000000000000*x + 4.00000000000000 over Complex Field with 53 bits of precision
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 2.00000000000000 over Real Field with 53 bits of precision
Real Field with 53 bits of precision
sage: E = EllipticCurve(QQbar,[5,6]); E; E.base_field()
Elliptic Curve defined by y^2 = x^3 + 5*x + 6 over Algebraic Field
Algebraic Field
See :trac:`6657` ::
sage: EllipticCurve(3,j=1728)
Traceback (most recent call last):
...
ValueError: First parameter (if present) must be a ring when j is specified
sage: EllipticCurve(GF(5),j=3/5)
Traceback (most recent call last):
...
ValueError: First parameter must be a ring containing 3/5
If the universe of the coefficients is a general field, the object
constructed has type EllipticCurve_field. Otherwise it is
EllipticCurve_generic. See :trac:`9816` ::
sage: E = EllipticCurve([QQbar(1),3]); E
Elliptic Curve defined by y^2 = x^3 + x + 3 over Algebraic Field
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_field.EllipticCurve_field_with_category'>
sage: E = EllipticCurve([RR(1),3]); E
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 3.00000000000000 over Real Field with 53 bits of precision
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_field.EllipticCurve_field_with_category'>
sage: E = EllipticCurve([i,i]); E
Elliptic Curve defined by y^2 = x^3 + I*x + I over Symbolic Ring
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_field.EllipticCurve_field_with_category'>
sage: E.category()
Category of schemes over Symbolic Ring
sage: SR in Fields()
True
sage: F = FractionField(PolynomialRing(QQ,'t'))
sage: t = F.gen()
sage: E = EllipticCurve([t,0]); E
Elliptic Curve defined by y^2 = x^3 + t*x over Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_field.EllipticCurve_field_with_category'>
sage: E.category()
Category of schemes over Fraction Field of Univariate Polynomial Ring in t over Rational Field
See :trac:`12517`::
sage: E = EllipticCurve([1..5])
sage: EllipticCurve(E.a_invariants())
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field
See :trac:`11773`::
sage: E = EllipticCurve()
Traceback (most recent call last):
...
TypeError: invalid input to EllipticCurve constructor
"""
import ell_generic, ell_field, ell_finite_field, ell_number_field, ell_rational_field, ell_padic_field # here to avoid circular includes
if j is not None:
if not x is None:
if is_Ring(x):
try:
j = x(j)
except (ZeroDivisionError, ValueError, TypeError):
raise ValueError, "First parameter must be a ring containing %s"%j
else:
raise ValueError, "First parameter (if present) must be a ring when j is specified"
return EllipticCurve_from_j(j, minimal_twist)
if x is None:
raise TypeError, "invalid input to EllipticCurve constructor"
if is_SymbolicEquation(x):
x = x.lhs() - x.rhs()
if parent(x) is SR:
x = x._polynomial_(rings.QQ['x', 'y'])
if is_MPolynomial(x):
if y is None:
return EllipticCurve_from_Weierstrass_polynomial(x)
else:
return EllipticCurve_from_cubic(x, y, morphism=False)
if is_Ring(x):
if is_RationalField(x):
return ell_rational_field.EllipticCurve_rational_field(x, y)
elif is_FiniteField(x) or (is_IntegerModRing(x) and x.characteristic().is_prime()):
return ell_finite_field.EllipticCurve_finite_field(x, y)
elif rings.is_pAdicField(x):
return ell_padic_field.EllipticCurve_padic_field(x, y)
elif is_NumberField(x):
return ell_number_field.EllipticCurve_number_field(x, y)
elif x in _Fields:
return ell_field.EllipticCurve_field(x, y)
return ell_generic.EllipticCurve_generic(x, y)
if isinstance(x, unicode):
x = str(x)
if isinstance(x, basestring):
return ell_rational_field.EllipticCurve_rational_field(x)
if is_RingElement(x) and y is None:
raise TypeError, "invalid input to EllipticCurve constructor"
if not isinstance(x, (list, tuple)):
raise TypeError, "invalid input to EllipticCurve constructor"
x = Sequence(x)
if not (len(x) in [2,5]):
raise ValueError, "sequence of coefficients must have length 2 or 5"
R = x.universe()
if isinstance(x[0], (rings.Rational, rings.Integer, int, long)):
return ell_rational_field.EllipticCurve_rational_field(x, y)
elif is_NumberField(R):
return ell_number_field.EllipticCurve_number_field(x, y)
elif rings.is_pAdicField(R):
return ell_padic_field.EllipticCurve_padic_field(x, y)
elif is_FiniteField(R) or (is_IntegerModRing(R) and R.characteristic().is_prime()):
return ell_finite_field.EllipticCurve_finite_field(x, y)
elif R in _Fields:
return ell_field.EllipticCurve_field(x, y)
return ell_generic.EllipticCurve_generic(x, y)
def _element_constructor_(self, x):
r"""
Try to turn ``x`` into an element of ``self``.
INPUT:
- ``x`` - an element of some group algebra or of a
ring or of a group
OUTPUT: ``x`` as a member of ``self``.
sage: G = KleinFourGroup()
sage: f = G.gen(0)
sage: ZG = GroupAlgebra(G)
sage: ZG(f) # indirect doctest
(3,4)
sage: ZG(1) == ZG(G(1))
True
sage: G = AbelianGroup(1)
sage: ZG = GroupAlgebra(G)
sage: f = ZG.group().gen()
sage: ZG(FormalSum([(1,f), (2, f**2)]))
f + 2*f^2
sage: G = GL(2,7)
sage: OG = GroupAlgebra(G, ZZ[sqrt(5)])
sage: OG(2)
2*[1 0]
[0 1]
sage: OG(G(2)) # conversion is not the obvious one
[2 0]
[0 2]
sage: OG(FormalSum([ (1, G(2)), (2, RR(0.77)) ]) )
Traceback (most recent call last):
...
TypeError: Attempt to coerce non-integral RealNumber to Integer
sage: OG(OG.base_ring().basis()[1])
sqrt5*[1 0]
[0 1]
"""
from sage.rings.ring import is_Ring
from sage.groups.group import is_Group
from sage.structure.formal_sum import FormalSum
k = self.base_ring()
G = self.group()
S = x.parent()
if isinstance(S, GroupAlgebra):
if self.has_coerce_map_from(S):
# coerce monomials, coerce coefficients, reassemble
d = x.monomial_coefficients()
new_d = {}
for g in d:
g1 = G(g)
if g1 in new_d:
new_d[g1] += k(d[g]) + new_d[g1]
else:
new_d[g1] = k(d[g])
return self._from_dict(new_d)
elif is_Ring(S):
# coerce to multiple of identity element
return k(x) * self(1)
elif is_Group(S):
# Check whether group coerces to base_ring first.
if k.has_coerce_map_from(S):
return k(x) * self(1)
if G.has_coerce_map_from(S):
return self.monomial(self.group()(x))
elif isinstance(x, FormalSum) and k.has_coerce_map_from(S.base_ring()):
y = [(G(g), k(coeff)) for coeff,g in x]
return self.sum_of_terms(y)
raise TypeError("Don't know how to create an element of %s from %s" % \
(self, x))
def random_quadraticform(R, n, rand_arg_list=[]):
"""
Create a random quadratic form in `n` variables defined over the ring `R`.
The last (and optional) argument ``rand_arg_list`` is a list of at most 3
elements which is passed (as at most 3 separate variables) into the method
``R.random_element()``.
INPUT:
- `R` -- a ring.
- `n` -- an integer `\ge 0`
- ``rand_arg_list`` -- a list of at most 3 arguments which can be taken by
``R.random_element()``.
OUTPUT:
A quadratic form over the ring `R`.
EXAMPLES::
sage: random_quadraticform(ZZ, 3, [1,5]) ## RANDOM
Quadratic form in 3 variables over Integer Ring with coefficients:
[ 3 2 3 ]
[ * 1 4 ]
[ * * 3 ]
::
sage: random_quadraticform(ZZ, 3, [-5,5]) ## RANDOM
Quadratic form in 3 variables over Integer Ring with coefficients:
[ 3 2 -5 ]
[ * 2 -2 ]
[ * * -5 ]
::
sage: random_quadraticform(ZZ, 3, [-50,50]) ## RANDOM
Quadratic form in 3 variables over Integer Ring with coefficients:
[ 1 8 -23 ]
[ * 0 0 ]
[ * * 6 ]
"""
## Sanity Checks: We have a ring and there are at most 3 parameters for randomness!
if len(rand_arg_list) > 3:
raise TypeError(
"Oops! The list of randomness arguments can have at most 3 elements."
)
if not is_Ring(R):
raise TypeError("Oops! The first argument must be a ring.")
## Create a list of upper-triangular entries for the quadratic form
L = len(rand_arg_list)
nn = int(n * (n + 1) / 2)
if L == 0:
rand_list = [R.random_element() for _ in range(nn)]
elif L == 1:
rand_list = [R.random_element(rand_arg_list[0]) for _ in range(nn)]
elif L == 2:
rand_list = [
R.random_element(rand_arg_list[0], rand_arg_list[1])
for _ in range(nn)
]
elif L == 3:
rand_list = [
R.random_element(rand_arg_list[0], rand_arg_list[1],
rand_arg_list[2]) for _ in range(nn)
]
## Return the Quadratic Form
return QuadraticForm(R, n, rand_list)
def random_quadraticform(R, n, rand_arg_list=[]):
"""
Create a random quadratic form in `n` variables defined over the ring `R`.
The last (and optional) argument ``rand_arg_list`` is a list of at most 3
elements which is passed (as at most 3 separate variables) into the method
``R.random_element()``.
INPUT:
- `R` -- a ring.
- `n` -- an integer `\ge 0`
- ``rand_arg_list`` -- a list of at most 3 arguments which can be taken by
``R.random_element()``.
OUTPUT:
A quadratic form over the ring `R`.
EXAMPLES::
sage: random_quadraticform(ZZ, 3, [1,5]) ## RANDOM
Quadratic form in 3 variables over Integer Ring with coefficients:
[ 3 2 3 ]
[ * 1 4 ]
[ * * 3 ]
::
sage: random_quadraticform(ZZ, 3, [-5,5]) ## RANDOM
Quadratic form in 3 variables over Integer Ring with coefficients:
[ 3 2 -5 ]
[ * 2 -2 ]
[ * * -5 ]
::
sage: random_quadraticform(ZZ, 3, [-50,50]) ## RANDOM
Quadratic form in 3 variables over Integer Ring with coefficients:
[ 1 8 -23 ]
[ * 0 0 ]
[ * * 6 ]
"""
## Sanity Checks: We have a ring and there are at most 3 parameters for randomness!
if len(rand_arg_list) > 3:
raise TypeError, "Oops! The list of randomness arguments can have at most 3 elements."
if not is_Ring(R):
raise TypeError, "Oops! The first argument must be a ring."
## Create a list of upper-triangular entries for the quadratic form
L = len(rand_arg_list)
nn = int(n*(n+1)/2)
if L == 0:
rand_list = [R.random_element() for _ in range(nn)]
elif L == 1:
rand_list = [R.random_element(rand_arg_list[0]) for _ in range(nn)]
elif L == 2:
rand_list = [R.random_element(rand_arg_list[0], rand_arg_list[1]) for _ in range(nn)]
elif L == 3:
rand_list = [R.random_element(rand_arg_list[0], rand_arg_list[1], rand_arg_list[2]) for _ in range(nn)]
## Return the Quadratic Form
return QuadraticForm(R, n, rand_list)
def LaurentPolynomialRing(base_ring, arg1=None, arg2=None, sparse = False, order='degrevlex', names = None, name=None):
r"""
Return the globally unique univariate or multivariate Laurent polynomial
ring with given properties and variable name or names.
There are four ways to call the Laurent polynomial ring constructor:
1. ``LaurentPolynomialRing(base_ring, name, sparse=False)``
2. ``LaurentPolynomialRing(base_ring, names, order='degrevlex')``
3. ``LaurentPolynomialRing(base_ring, name, n, order='degrevlex')``
4. ``LaurentPolynomialRing(base_ring, n, name, order='degrevlex')``
The optional arguments sparse and order *must* be explicitly
named, and the other arguments must be given positionally.
INPUT:
- ``base_ring`` -- a commutative ring
- ``name`` -- a string
- ``names`` -- a list or tuple of names, or a comma separated string
- ``n`` -- a positive integer
- ``sparse`` -- bool (default: False), whether or not elements are sparse
- ``order`` -- string or
:class:`~sage.rings.polynomial.term_order.TermOrder`, e.g.,
- ``'degrevlex'`` (default) -- degree reverse lexicographic
- ``'lex'`` -- lexicographic
- ``'deglex'`` -- degree lexicographic
- ``TermOrder('deglex',3) + TermOrder('deglex',3)`` -- block ordering
OUTPUT:
``LaurentPolynomialRing(base_ring, name, sparse=False)`` returns a
univariate Laurent polynomial ring; all other input formats return a
multivariate Laurent polynomial ring.
UNIQUENESS and IMMUTABILITY: In Sage there is exactly one
single-variate Laurent polynomial ring over each base ring in each choice
of variable and sparseness. There is also exactly one multivariate
Laurent polynomial ring over each base ring for each choice of names of
variables and term order.
::
sage: R.<x,y> = LaurentPolynomialRing(QQ,2); R
Multivariate Laurent Polynomial Ring in x, y over Rational Field
sage: f = x^2 - 2*y^-2
You can't just globally change the names of those variables.
This is because objects all over Sage could have pointers to
that polynomial ring.
::
sage: R._assign_names(['z','w'])
Traceback (most recent call last):
...
ValueError: variable names cannot be changed after object creation.
EXAMPLES:
1. ``LaurentPolynomialRing(base_ring, name, sparse=False)``
::
sage: LaurentPolynomialRing(QQ, 'w')
Univariate Laurent Polynomial Ring in w over Rational Field
Use the diamond brackets notation to make the variable
ready for use after you define the ring::
sage: R.<w> = LaurentPolynomialRing(QQ)
sage: (1 + w)^3
1 + 3*w + 3*w^2 + w^3
You must specify a name::
sage: LaurentPolynomialRing(QQ)
Traceback (most recent call last):
...
TypeError: You must specify the names of the variables.
sage: R.<abc> = LaurentPolynomialRing(QQ, sparse=True); R
Univariate Laurent Polynomial Ring in abc over Rational Field
sage: R.<w> = LaurentPolynomialRing(PolynomialRing(GF(7),'k')); R
Univariate Laurent Polynomial Ring in w over Univariate Polynomial Ring in k over Finite Field of size 7
Rings with different variables are different::
sage: LaurentPolynomialRing(QQ, 'x') == LaurentPolynomialRing(QQ, 'y')
False
2. ``LaurentPolynomialRing(base_ring, names, order='degrevlex')``
::
sage: R = LaurentPolynomialRing(QQ, 'a,b,c'); R
Multivariate Laurent Polynomial Ring in a, b, c over Rational Field
sage: S = LaurentPolynomialRing(QQ, ['a','b','c']); S
Multivariate Laurent Polynomial Ring in a, b, c over Rational Field
sage: T = LaurentPolynomialRing(QQ, ('a','b','c')); T
Multivariate Laurent Polynomial Ring in a, b, c over Rational Field
All three rings are identical.
::
sage: (R is S) and (S is T)
True
There is a unique Laurent polynomial ring with each term order::
sage: R = LaurentPolynomialRing(QQ, 'x,y,z', order='degrevlex'); R
Multivariate Laurent Polynomial Ring in x, y, z over Rational Field
sage: S = LaurentPolynomialRing(QQ, 'x,y,z', order='invlex'); S
Multivariate Laurent Polynomial Ring in x, y, z over Rational Field
sage: S is LaurentPolynomialRing(QQ, 'x,y,z', order='invlex')
True
sage: R == S
False
3. ``LaurentPolynomialRing(base_ring, name, n, order='degrevlex')``
If you specify a single name as a string and a number of
variables, then variables labeled with numbers are created.
::
sage: LaurentPolynomialRing(QQ, 'x', 10)
Multivariate Laurent Polynomial Ring in x0, x1, x2, x3, x4, x5, x6, x7, x8, x9 over Rational Field
sage: LaurentPolynomialRing(GF(7), 'y', 5)
Multivariate Laurent Polynomial Ring in y0, y1, y2, y3, y4 over Finite Field of size 7
sage: LaurentPolynomialRing(QQ, 'y', 3, sparse=True)
Multivariate Laurent Polynomial Ring in y0, y1, y2 over Rational Field
By calling the
:meth:`~sage.structure.category_object.CategoryObject.inject_variables`
method, all those variable names are available for interactive use::
sage: R = LaurentPolynomialRing(GF(7),15,'w'); R
Multivariate Laurent Polynomial Ring in w0, w1, w2, w3, w4, w5, w6, w7, w8, w9, w10, w11, w12, w13, w14 over Finite Field of size 7
sage: R.inject_variables()
Defining w0, w1, w2, w3, w4, w5, w6, w7, w8, w9, w10, w11, w12, w13, w14
sage: (w0 + 2*w8 + w13)^2
w0^2 + 4*w0*w8 + 4*w8^2 + 2*w0*w13 + 4*w8*w13 + w13^2
"""
if is_Element(arg1) and not isinstance(arg1, (int, long, Integer)):
arg1 = repr(arg1)
if is_Element(arg2) and not isinstance(arg2, (int, long, Integer)):
arg2 = repr(arg2)
if isinstance(arg1, (int, long, Integer)):
arg1, arg2 = arg2, arg1
if not names is None:
arg1 = names
elif not name is None:
arg1 = name
if not is_Ring(base_ring):
raise TypeError('base_ring must be a ring')
if arg1 is None:
raise TypeError("You must specify the names of the variables.")
R = None
if isinstance(arg1, (list, tuple)):
arg1 = [str(x) for x in arg1]
if isinstance(arg2, (list, tuple)):
arg2 = [str(x) for x in arg2]
if isinstance(arg2, (int, long, Integer)):
# 3. LaurentPolynomialRing(base_ring, names, n, order='degrevlex'):
if not isinstance(arg1, (list, tuple, str)):
raise TypeError("You *must* specify the names of the variables.")
n = int(arg2)
names = arg1
R = _multi_variate(base_ring, names, n, sparse, order)
elif isinstance(arg1, str) or (isinstance(arg1, (list,tuple)) and len(arg1) == 1) and isinstance(arg1[0], str):
if isinstance(arg1, (list,tuple)):
arg1 = arg1[0]
if not ',' in arg1:
# 1. LaurentPolynomialRing(base_ring, name, sparse=False):
if not arg2 is None:
raise TypeError("if second arguments is a string with no commas, then there must be no other non-optional arguments")
name = arg1
R = _single_variate(base_ring, name, sparse)
else:
# 2-4. LaurentPolynomialRing(base_ring, names, order='degrevlex'):
if not arg2 is None:
raise TypeError("invalid input to LaurentPolynomialRing function; please see the docstring for that function")
names = arg1.split(',')
n = len(names)
R = _multi_variate(base_ring, names, n, sparse, order)
elif isinstance(arg1, (list, tuple)):
# LaurentPolynomialRing(base_ring, names (list or tuple), order='degrevlex'):
names = arg1
n = len(names)
R = _multi_variate(base_ring, names, n, sparse, order)
if arg1 is None and arg2 is None:
raise TypeError("you *must* specify the indeterminates (as not None).")
if R is None:
raise TypeError("invalid input (%s, %s, %s) to PolynomialRing function; please see the docstring for that function"%(base_ring, arg1, arg2))
return R
def create_key_and_extra_args(self, x=None, y=None, j=None, minimal_twist=True, **kwds):
"""
Return a ``UniqueFactory`` key and possibly extra parameters.
INPUT:
See the documentation for :class:`EllipticCurveFactory`.
OUTPUT:
A pair ``(key, extra_args)``:
- ``key`` has the form `(R, (a_1, a_2, a_3, a_4, a_6))`,
representing a ring and the Weierstrass coefficients of an
elliptic curve over that ring;
- ``extra_args`` is a dictionary containing additional data to
be inserted into the elliptic curve structure.
EXAMPLES::
sage: EllipticCurve.create_key_and_extra_args(j=8000)
((Rational Field, (0, -1, 0, -3, -1)), {})
When constructing a curve over `\\QQ` from a Cremona or LMFDB
label, the invariants from the database are returned as
``extra_args``::
sage: key, data = EllipticCurve.create_key_and_extra_args('389.a1')
sage: key
(Rational Field, (0, 1, 1, -2, 0))
sage: data['conductor']
389
sage: data['cremona_label']
'389a1'
sage: data['lmfdb_label']
'389.a1'
sage: data['rank']
2
sage: data['torsion_order']
1
User-specified keywords are also included in ``extra_args``::
sage: key, data = EllipticCurve.create_key_and_extra_args((0, 0, 1, -23737, 960366), rank=4)
sage: data['rank']
4
Furthermore, keywords takes precedence over data from the
database, which can be used to specify an alternative set of
generators for the Mordell-Weil group::
sage: key, data = EllipticCurve.create_key_and_extra_args('5077a1', gens=[[1, -1], [-2, 3], [4, -7]])
sage: data['gens']
[[1, -1], [-2, 3], [4, -7]]
sage: E = EllipticCurve.create_object(0, key, **data)
sage: E.gens()
[(-2 : 3 : 1), (1 : -1 : 1), (4 : -7 : 1)]
Note that elliptic curves are equal if and only they have the
same base ring and Weierstrass equation; the data in
``extra_args`` do not influence comparison of elliptic curves.
A consequence of this is that passing keyword arguments only
works when constructing an elliptic curve the first time:
sage: E = EllipticCurve('433a1', gens=[[-1, 1], [3, 4]])
sage: E.gens()
[(-1 : 1 : 1), (3 : 4 : 1)]
sage: E = EllipticCurve('433a1', gens=[[-1, 0], [0, 1]])
sage: E.gens()
[(-1 : 1 : 1), (3 : 4 : 1)]
.. WARNING::
Manually specifying extra data is almost never necessary
and is not guaranteed to have any effect, as the above
example shows. Almost no checking is done, so specifying
incorrect data may lead to wrong results of computations
instead of errors or warnings.
"""
R = None
if is_Ring(x):
(R, x) = (x, y)
if j is not None:
if R is not None:
try:
j = R(j)
except (ZeroDivisionError, ValueError, TypeError):
raise ValueError("First parameter must be a ring containing %s" % j)
elif x is not None:
raise ValueError("First parameter (if present) must be a ring when j is specified")
x = coefficients_from_j(j, minimal_twist)
if is_SymbolicEquation(x):
x = x.lhs() - x.rhs()
if parent(x) is SR:
x = x._polynomial_(rings.QQ['x', 'y'])
if is_MPolynomial(x):
if y is None:
x = coefficients_from_Weierstrass_polynomial(x)
else:
x = coefficients_from_cubic(x, y, morphism=False)
if isinstance(x, basestring):
# Interpret x as a Cremona or LMFDB label.
from sage.databases.cremona import CremonaDatabase
x, data = CremonaDatabase().coefficients_and_data(x)
# User-provided keywords may override database entries.
data.update(kwds)
kwds = data
if not isinstance(x, (list, tuple)):
raise TypeError("invalid input to EllipticCurve constructor")
if len(x) == 2:
x = (0, 0, 0, x[0], x[1])
elif len(x) != 5:
raise ValueError("sequence of coefficients must have length 2 or 5")
if R is None:
R = Sequence(x).universe()
if R in (rings.ZZ, int, long):
R = rings.QQ
return (R, tuple(R(a) for a in x)), kwds
def create_key_and_extra_args(self,
x=None,
y=None,
j=None,
minimal_twist=True,
**kwds):
"""
Return a ``UniqueFactory`` key and possibly extra parameters.
INPUT:
See the documentation for :class:`EllipticCurveFactory`.
OUTPUT:
A pair ``(key, extra_args)``:
- ``key`` has the form `(R, (a_1, a_2, a_3, a_4, a_6))`,
representing a ring and the Weierstrass coefficients of an
elliptic curve over that ring;
- ``extra_args`` is a dictionary containing additional data to
be inserted into the elliptic curve structure.
EXAMPLES::
sage: EllipticCurve.create_key_and_extra_args(j=8000)
((Rational Field, (0, -1, 0, -3, -1)), {})
When constructing a curve over `\\QQ` from a Cremona or LMFDB
label, the invariants from the database are returned as
``extra_args``::
sage: key, data = EllipticCurve.create_key_and_extra_args('389.a1')
sage: key
(Rational Field, (0, 1, 1, -2, 0))
sage: data['conductor']
389
sage: data['cremona_label']
'389a1'
sage: data['lmfdb_label']
'389.a1'
sage: data['rank']
2
sage: data['torsion_order']
1
User-specified keywords are also included in ``extra_args``::
sage: key, data = EllipticCurve.create_key_and_extra_args((0, 0, 1, -23737, 960366), rank=4)
sage: data['rank']
4
Furthermore, keywords takes precedence over data from the
database, which can be used to specify an alternative set of
generators for the Mordell-Weil group::
sage: key, data = EllipticCurve.create_key_and_extra_args('5077a1', gens=[[1, -1], [-2, 3], [4, -7]])
sage: data['gens']
[[1, -1], [-2, 3], [4, -7]]
sage: E = EllipticCurve.create_object(0, key, **data)
sage: E.gens()
[(-2 : 3 : 1), (1 : -1 : 1), (4 : -7 : 1)]
Note that elliptic curves are equal if and only they have the
same base ring and Weierstrass equation; the data in
``extra_args`` do not influence comparison of elliptic curves.
A consequence of this is that passing keyword arguments only
works when constructing an elliptic curve the first time::
sage: E = EllipticCurve('433a1', gens=[[-1, 1], [3, 4]])
sage: E.gens()
[(-1 : 1 : 1), (3 : 4 : 1)]
sage: E = EllipticCurve('433a1', gens=[[-1, 0], [0, 1]])
sage: E.gens()
[(-1 : 1 : 1), (3 : 4 : 1)]
.. WARNING::
Manually specifying extra data is almost never necessary
and is not guaranteed to have any effect, as the above
example shows. Almost no checking is done, so specifying
incorrect data may lead to wrong results of computations
instead of errors or warnings.
"""
R = None
if is_Ring(x):
(R, x) = (x, y)
if j is not None:
if R is not None:
try:
j = R(j)
except (ZeroDivisionError, ValueError, TypeError):
raise ValueError(
"First parameter must be a ring containing %s" % j)
elif x is not None:
raise ValueError(
"First parameter (if present) must be a ring when j is specified"
)
x = coefficients_from_j(j, minimal_twist)
if is_SymbolicEquation(x):
x = x.lhs() - x.rhs()
if parent(x) is SR:
x = x._polynomial_(rings.QQ['x', 'y'])
if is_MPolynomial(x):
if y is None:
x = coefficients_from_Weierstrass_polynomial(x)
else:
x = coefficients_from_cubic(x, y, morphism=False)
if isinstance(x, string_types):
# Interpret x as a Cremona or LMFDB label.
from sage.databases.cremona import CremonaDatabase
x, data = CremonaDatabase().coefficients_and_data(x)
# User-provided keywords may override database entries.
data.update(kwds)
kwds = data
if not isinstance(x, (list, tuple)):
raise TypeError("invalid input to EllipticCurve constructor")
if len(x) == 2:
x = (0, 0, 0, x[0], x[1])
elif len(x) != 5:
raise ValueError(
"sequence of coefficients must have length 2 or 5")
if R is None:
R = Sequence(x).universe()
if R in (rings.ZZ, ) + integer_types:
R = rings.QQ
return (R, tuple(R(a) for a in x)), kwds
def EllipticCurve(x=None, y=None, j=None, minimal_twist=True):
r"""
Construct an elliptic curve.
In Sage, an elliptic curve is always specified by its a-invariants
.. math::
y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.
INPUT:
There are several ways to construct an elliptic curve:
- ``EllipticCurve([a1,a2,a3,a4,a6])``: Elliptic curve with given
a-invariants. The invariants are coerced into the parent of the
first element. If all are integers, they are coerced into the
rational numbers.
- ``EllipticCurve([a4,a6])``: Same as above, but `a_1=a_2=a_3=0`.
- ``EllipticCurve(label)``: Returns the elliptic curve over Q from
the Cremona database with the given label. The label is a
string, such as ``"11a"`` or ``"37b2"``. The letters in the
label *must* be lower case (Cremona's new labeling).
- ``EllipticCurve(R, [a1,a2,a3,a4,a6])``: Create the elliptic
curve over ``R`` with given a-invariants. Here ``R`` can be an
arbitrary ring. Note that addition need not be defined.
- ``EllipticCurve(j=j0)`` or ``EllipticCurve_from_j(j0)``: Return
an elliptic curve with j-invariant ``j0``.
- ``EllipticCurve(polynomial)``: Read off the a-invariants from
the polynomial coefficients, see
:func:`EllipticCurve_from_Weierstrass_polynomial`.
In each case above where the input is a list of length 2 or 5, one
can instead give a 2 or 5-tuple instead.
EXAMPLES:
We illustrate creating elliptic curves::
sage: EllipticCurve([0,0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
We create a curve from a Cremona label::
sage: EllipticCurve('37b2')
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 1873*x - 31833 over Rational Field
sage: EllipticCurve('5077a')
Elliptic Curve defined by y^2 + y = x^3 - 7*x + 6 over Rational Field
sage: EllipticCurve('389a')
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
Old Cremona labels are allowed::
sage: EllipticCurve('2400FF')
Elliptic Curve defined by y^2 = x^3 + x^2 + 2*x + 8 over Rational Field
Unicode labels are allowed::
sage: EllipticCurve(u'389a')
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
We create curves over a finite field as follows::
sage: EllipticCurve([GF(5)(0),0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5
sage: EllipticCurve(GF(5), [0, 0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5
Elliptic curves over `\ZZ/N\ZZ` with `N` prime are of type
"elliptic curve over a finite field"::
sage: F = Zmod(101)
sage: EllipticCurve(F, [2, 3])
Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Ring of integers modulo 101
sage: E = EllipticCurve([F(2), F(3)])
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_finite_field.EllipticCurve_finite_field_with_category'>
sage: E.category()
Category of schemes over Ring of integers modulo 101
In contrast, elliptic curves over `\ZZ/N\ZZ` with `N` composite
are of type "generic elliptic curve"::
sage: F = Zmod(95)
sage: EllipticCurve(F, [2, 3])
Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Ring of integers modulo 95
sage: E = EllipticCurve([F(2), F(3)])
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_generic.EllipticCurve_generic_with_category'>
sage: E.category()
Category of schemes over Ring of integers modulo 95
The following is a curve over the complex numbers::
sage: E = EllipticCurve(CC, [0,0,1,-1,0])
sage: E
Elliptic Curve defined by y^2 + 1.00000000000000*y = x^3 + (-1.00000000000000)*x over Complex Field with 53 bits of precision
sage: E.j_invariant()
2988.97297297297
We can also create elliptic curves by giving the Weierstrass equation::
sage: x, y = var('x,y')
sage: EllipticCurve(y^2 + y == x^3 + x - 9)
Elliptic Curve defined by y^2 + y = x^3 + x - 9 over Rational Field
sage: R.<x,y> = GF(5)[]
sage: EllipticCurve(x^3 + x^2 + 2 - y^2 - y*x)
Elliptic Curve defined by y^2 + x*y = x^3 + x^2 + 2 over Finite Field of size 5
We can explicitly specify the `j`-invariant::
sage: E = EllipticCurve(j=1728); E; E.j_invariant(); E.label()
Elliptic Curve defined by y^2 = x^3 - x over Rational Field
1728
'32a2'
sage: E = EllipticCurve(j=GF(5)(2)); E; E.j_invariant()
Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 5
2
See :trac:`6657` ::
sage: EllipticCurve(GF(144169),j=1728)
Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 144169
By default, when a rational value of `j` is given, the constructed
curve is a minimal twist (minimal conductor for curves with that
`j`-invariant). This can be changed by setting the optional
parameter ``minimal_twist``, which is True by default, to False::
sage: EllipticCurve(j=100)
Elliptic Curve defined by y^2 = x^3 + x^2 + 3392*x + 307888 over Rational Field
sage: E =EllipticCurve(j=100); E
Elliptic Curve defined by y^2 = x^3 + x^2 + 3392*x + 307888 over Rational Field
sage: E.conductor()
33129800
sage: E.j_invariant()
100
sage: E =EllipticCurve(j=100, minimal_twist=False); E
Elliptic Curve defined by y^2 = x^3 + 488400*x - 530076800 over Rational Field
sage: E.conductor()
298168200
sage: E.j_invariant()
100
Without this option, constructing the curve could take a long time
since both `j` and `j-1728` have to be factored to compute the
minimal twist (see :trac:`13100`)::
sage: E = EllipticCurve_from_j(2^256+1,minimal_twist=False)
sage: E.j_invariant() == 2^256+1
True
TESTS::
sage: R = ZZ['u', 'v']
sage: EllipticCurve(R, [1,1])
Elliptic Curve defined by y^2 = x^3 + x + 1 over Multivariate Polynomial Ring in u, v
over Integer Ring
We create a curve and a point over QQbar (see #6879)::
sage: E = EllipticCurve(QQbar,[0,1])
sage: E(0)
(0 : 1 : 0)
sage: E.base_field()
Algebraic Field
sage: E = EllipticCurve(RR,[1,2]); E; E.base_field()
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 2.00000000000000 over Real Field with 53 bits of precision
Real Field with 53 bits of precision
sage: EllipticCurve(CC,[3,4]); E; E.base_field()
Elliptic Curve defined by y^2 = x^3 + 3.00000000000000*x + 4.00000000000000 over Complex Field with 53 bits of precision
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 2.00000000000000 over Real Field with 53 bits of precision
Real Field with 53 bits of precision
sage: E = EllipticCurve(QQbar,[5,6]); E; E.base_field()
Elliptic Curve defined by y^2 = x^3 + 5*x + 6 over Algebraic Field
Algebraic Field
See :trac:`6657` ::
sage: EllipticCurve(3,j=1728)
Traceback (most recent call last):
...
ValueError: First parameter (if present) must be a ring when j is specified
sage: EllipticCurve(GF(5),j=3/5)
Traceback (most recent call last):
...
ValueError: First parameter must be a ring containing 3/5
If the universe of the coefficients is a general field, the object
constructed has type EllipticCurve_field. Otherwise it is
EllipticCurve_generic. See :trac:`9816` ::
sage: E = EllipticCurve([QQbar(1),3]); E
Elliptic Curve defined by y^2 = x^3 + x + 3 over Algebraic Field
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_field.EllipticCurve_field_with_category'>
sage: E = EllipticCurve([RR(1),3]); E
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 3.00000000000000 over Real Field with 53 bits of precision
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_field.EllipticCurve_field_with_category'>
sage: E = EllipticCurve([i,i]); E
Elliptic Curve defined by y^2 = x^3 + I*x + I over Symbolic Ring
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_field.EllipticCurve_field_with_category'>
sage: E.category()
Category of schemes over Symbolic Ring
sage: SR in Fields()
True
sage: F = FractionField(PolynomialRing(QQ,'t'))
sage: t = F.gen()
sage: E = EllipticCurve([t,0]); E
Elliptic Curve defined by y^2 = x^3 + t*x over Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_field.EllipticCurve_field_with_category'>
sage: E.category()
Category of schemes over Fraction Field of Univariate Polynomial Ring in t over Rational Field
See :trac:`12517`::
sage: E = EllipticCurve([1..5])
sage: EllipticCurve(E.a_invariants())
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field
See :trac:`11773`::
sage: E = EllipticCurve()
Traceback (most recent call last):
...
TypeError: invalid input to EllipticCurve constructor
"""
import ell_generic, ell_field, ell_finite_field, ell_number_field, ell_rational_field, ell_padic_field # here to avoid circular includes
if j is not None:
if not x is None:
if is_Ring(x):
try:
j = x(j)
except (ZeroDivisionError, ValueError, TypeError):
raise ValueError, "First parameter must be a ring containing %s" % j
else:
raise ValueError, "First parameter (if present) must be a ring when j is specified"
return EllipticCurve_from_j(j, minimal_twist)
if x is None:
raise TypeError, "invalid input to EllipticCurve constructor"
if is_SymbolicEquation(x):
x = x.lhs() - x.rhs()
if parent(x) is SR:
x = x._polynomial_(rings.QQ['x', 'y'])
if is_MPolynomial(x):
if y is None:
return EllipticCurve_from_Weierstrass_polynomial(x)
else:
return EllipticCurve_from_cubic(x, y, morphism=False)
if is_Ring(x):
if is_RationalField(x):
return ell_rational_field.EllipticCurve_rational_field(x, y)
elif is_FiniteField(x) or (is_IntegerModRing(x)
and x.characteristic().is_prime()):
return ell_finite_field.EllipticCurve_finite_field(x, y)
elif rings.is_pAdicField(x):
return ell_padic_field.EllipticCurve_padic_field(x, y)
elif is_NumberField(x):
return ell_number_field.EllipticCurve_number_field(x, y)
elif x in _Fields:
return ell_field.EllipticCurve_field(x, y)
return ell_generic.EllipticCurve_generic(x, y)
if isinstance(x, unicode):
x = str(x)
if isinstance(x, basestring):
return ell_rational_field.EllipticCurve_rational_field(x)
if is_RingElement(x) and y is None:
raise TypeError, "invalid input to EllipticCurve constructor"
if not isinstance(x, (list, tuple)):
raise TypeError, "invalid input to EllipticCurve constructor"
x = Sequence(x)
if not (len(x) in [2, 5]):
raise ValueError, "sequence of coefficients must have length 2 or 5"
R = x.universe()
if isinstance(x[0], (rings.Rational, rings.Integer, int, long)):
return ell_rational_field.EllipticCurve_rational_field(x, y)
elif is_NumberField(R):
return ell_number_field.EllipticCurve_number_field(x, y)
elif rings.is_pAdicField(R):
return ell_padic_field.EllipticCurve_padic_field(x, y)
elif is_FiniteField(R) or (is_IntegerModRing(R)
and R.characteristic().is_prime()):
return ell_finite_field.EllipticCurve_finite_field(x, y)
elif R in _Fields:
return ell_field.EllipticCurve_field(x, y)
return ell_generic.EllipticCurve_generic(x, y)
