def __init__(self, E=None, urst=None, F=None):
r"""
Constructor for WeierstrassIsomorphism class,
INPUT:
- ``E`` -- an EllipticCurve, or None (see below).
- ``urst`` -- a 4-tuple `(u,r,s,t)`, or None (see below).
- ``F`` -- an EllipticCurve, or None (see below).
Given two Elliptic Curves ``E`` and ``F`` (represented by
Weierstrass models as usual), and a transformation ``urst``
from ``E`` to ``F``, construct an isomorphism from ``E`` to
``F``. An exception is raised if ``urst(E)!=F``. At most one
of ``E``, ``F``, ``urst`` can be None. If ``F==None`` then
``F`` is constructed as ``urst(E)``. If ``E==None`` then
``E`` is constructed as ``urst^-1(F)``. If ``urst==None``
then an isomorphism from ``E`` to ``F`` is constructed if
possible, and an exception is raised if they are not
isomorphic. Otherwise ``urst`` can be a tuple of length 4 or
a object of type ``baseWI``.
Users will not usually need to use this class directly, but instead use
methods such as ``isomorphism`` of elliptic curves.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.weierstrass_morphism import *
sage: WeierstrassIsomorphism(EllipticCurve([0,1,2,3,4]),(-1,2,3,4))
Generic morphism:
From: Abelian group of points on Elliptic Curve defined by y^2 + 2*y = x^3 + x^2 + 3*x + 4 over Rational Field
To: Abelian group of points on Elliptic Curve defined by y^2 - 6*x*y - 10*y = x^3 - 2*x^2 - 11*x - 2 over Rational Field
Via: (u,r,s,t) = (-1, 2, 3, 4)
sage: E=EllipticCurve([0,1,2,3,4])
sage: F=EllipticCurve(E.cremona_label())
sage: WeierstrassIsomorphism(E,None,F)
Generic morphism:
From: Abelian group of points on Elliptic Curve defined by y^2 + 2*y = x^3 + x^2 + 3*x + 4 over Rational Field
To: Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x^2 + 3*x + 5 over Rational Field
Via: (u,r,s,t) = (1, 0, 0, -1)
sage: w=WeierstrassIsomorphism(None,(1,0,0,-1),F)
sage: w._domain_curve==E
True
"""
from ell_generic import is_EllipticCurve
if E != None:
if not is_EllipticCurve(E):
raise ValueError, "First argument must be an elliptic curve or None"
if F != None:
if not is_EllipticCurve(F):
raise ValueError, "Third argument must be an elliptic curve or None"
if urst != None:
if len(urst) != 4:
raise ValueError, "Second argument must be [u,r,s,t] or None"
if len([par for par in [E, urst, F] if par != None]) < 2:
raise ValueError, "At most 1 argument can be None"
if F == None: # easy case
baseWI.__init__(self, *urst)
F = EllipticCurve(baseWI.__call__(self, list(E.a_invariants())))
Morphism.__init__(self, Hom(E(0).parent(), F(0).parent()))
self._domain_curve = E
self._codomain_curve = F
return
if E == None: # easy case in reverse
baseWI.__init__(self, *urst)
inv_urst = baseWI.__invert__(self)
E = EllipticCurve(baseWI.__call__(inv_urst,
list(F.a_invariants())))
Morphism.__init__(self, Hom(E(0).parent(), F(0).parent()))
self._domain_curve = E
self._codomain_curve = F
return
if urst == None: # try to construct the morphism
urst = isomorphisms(E, F, True)
if urst == None:
raise ValueError, "Elliptic curves not isomorphic."
baseWI.__init__(self, *urst)
Morphism.__init__(self, Hom(E(0).parent(), F(0).parent()))
self._domain_curve = E
self._codomain_curve = F
return
# none of the parameters is None:
baseWI.__init__(self, *urst)
if F != EllipticCurve(baseWI.__call__(self, list(E.a_invariants()))):
raise ValueError, "second argument is not an isomorphism from first argument to third argument"
else:
Morphism.__init__(self, Hom(E(0).parent(), F(0).parent()))
self._domain_curve = E
self._codomain_curve = F
return
def __init__(self, E=None, urst=None, F=None):
r"""
Constructor for WeierstrassIsomorphism class,
INPUT:
- ``E`` -- an EllipticCurve, or None (see below).
- ``urst`` -- a 4-tuple `(u,r,s,t)`, or None (see below).
- ``F`` -- an EllipticCurve, or None (see below).
Given two Elliptic Curves ``E`` and ``F`` (represented by
Weierstrass models as usual), and a transformation ``urst``
from ``E`` to ``F``, construct an isomorphism from ``E`` to
``F``. An exception is raised if ``urst(E)!=F``. At most one
of ``E``, ``F``, ``urst`` can be None. If ``F==None`` then
``F`` is constructed as ``urst(E)``. If ``E==None`` then
``E`` is constructed as ``urst^-1(F)``. If ``urst==None``
then an isomorphism from ``E`` to ``F`` is constructed if
possible, and an exception is raised if they are not
isomorphic. Otherwise ``urst`` can be a tuple of length 4 or
a object of type ``baseWI``.
Users will not usually need to use this class directly, but instead use
methods such as ``isomorphism`` of elliptic curves.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.weierstrass_morphism import *
sage: WeierstrassIsomorphism(EllipticCurve([0,1,2,3,4]),(-1,2,3,4))
Generic morphism:
From: Abelian group of points on Elliptic Curve defined by y^2 + 2*y = x^3 + x^2 + 3*x + 4 over Rational Field
To: Abelian group of points on Elliptic Curve defined by y^2 - 6*x*y - 10*y = x^3 - 2*x^2 - 11*x - 2 over Rational Field
Via: (u,r,s,t) = (-1, 2, 3, 4)
sage: E=EllipticCurve([0,1,2,3,4])
sage: F=EllipticCurve(E.cremona_label())
sage: WeierstrassIsomorphism(E,None,F)
Generic morphism:
From: Abelian group of points on Elliptic Curve defined by y^2 + 2*y = x^3 + x^2 + 3*x + 4 over Rational Field
To: Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x^2 + 3*x + 5 over Rational Field
Via: (u,r,s,t) = (1, 0, 0, -1)
sage: w=WeierstrassIsomorphism(None,(1,0,0,-1),F)
sage: w._domain_curve==E
True
"""
from ell_generic import is_EllipticCurve
if E!=None:
if not is_EllipticCurve(E):
raise ValueError("First argument must be an elliptic curve or None")
if F!=None:
if not is_EllipticCurve(F):
raise ValueError("Third argument must be an elliptic curve or None")
if urst!=None:
if len(urst)!=4:
raise ValueError("Second argument must be [u,r,s,t] or None")
if len([par for par in [E,urst,F] if par!=None])<2:
raise ValueError("At most 1 argument can be None")
if F==None: # easy case
baseWI.__init__(self,*urst)
F=EllipticCurve(baseWI.__call__(self,list(E.a_invariants())))
Morphism.__init__(self, Hom(E(0).parent(), F(0).parent()))
self._domain_curve = E
self._codomain_curve = F
return
if E==None: # easy case in reverse
baseWI.__init__(self,*urst)
inv_urst=baseWI.__invert__(self)
E=EllipticCurve(baseWI.__call__(inv_urst,list(F.a_invariants())))
Morphism.__init__(self, Hom(E(0).parent(), F(0).parent()))
self._domain_curve = E
self._codomain_curve = F
return
if urst==None: # try to construct the morphism
urst=isomorphisms(E,F,True)
if urst==None:
raise ValueError("Elliptic curves not isomorphic.")
baseWI.__init__(self, *urst)
Morphism.__init__(self, Hom(E(0).parent(), F(0).parent()))
self._domain_curve = E
self._codomain_curve = F
return
# none of the parameters is None:
baseWI.__init__(self,*urst)
if F!=EllipticCurve(baseWI.__call__(self,list(E.a_invariants()))):
raise ValueError("second argument is not an isomorphism from first argument to third argument")
else:
Morphism.__init__(self, Hom(E(0).parent(), F(0).parent()))
self._domain_curve = E
self._codomain_curve = F
return
def isomorphisms(E, F, JustOne=False):
r"""
Returns one or all isomorphisms between two elliptic curves.
INPUT:
- ``E``, ``F`` (EllipticCurve) -- Two elliptic curves.
- ``JustOne`` (bool) If True, returns one isomorphism, or None if
the curves are not isomorphic. If False, returns a (possibly
empty) list of isomorphisms.
OUTPUT:
Either None, or a 4-tuple `(u,r,s,t)` representing an isomorphism,
or a list of these.
.. note::
This function is not intended for users, who should use the
interface provided by ``ell_generic``.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.weierstrass_morphism import *
sage: isomorphisms(EllipticCurve_from_j(0),EllipticCurve('27a3'))
[(-1, 0, 0, -1), (1, 0, 0, 0)]
sage: isomorphisms(EllipticCurve_from_j(0),EllipticCurve('27a3'),JustOne=True)
(1, 0, 0, 0)
sage: isomorphisms(EllipticCurve_from_j(0),EllipticCurve('27a1'))
[]
sage: isomorphisms(EllipticCurve_from_j(0),EllipticCurve('27a1'),JustOne=True)
"""
from ell_generic import is_EllipticCurve
if not is_EllipticCurve(E) or not is_EllipticCurve(F):
raise ValueError, "arguments are not elliptic curves"
K = E.base_ring()
# if not K == F.base_ring(): return []
j = E.j_invariant()
if j != F.j_invariant():
if JustOne: return None
return []
from sage.rings.all import PolynomialRing
x = PolynomialRing(K, 'x').gen()
a1E, a2E, a3E, a4E, a6E = E.ainvs()
a1F, a2F, a3F, a4F, a6F = F.ainvs()
char = K.characteristic()
if char == 2:
if j == 0:
ulist = (x**3 - (a3E / a3F)).roots(multiplicities=False)
ans = []
for u in ulist:
slist = (x**4 + a3E * x + (a2F**2 + a4F) * u**4 + a2E**2 +
a4E).roots(multiplicities=False)
for s in slist:
r = s**2 + a2E + a2F * u**2
tlist = (x**2 + a3E * x + r**3 + a2E * r**2 + a4E * r +
a6E + a6F * u**6).roots(multiplicities=False)
for t in tlist:
if JustOne: return (u, r, s, t)
ans.append((u, r, s, t))
if JustOne: return None
ans.sort()
return ans
else:
ans = []
u = a1E / a1F
r = (a3E + a3F * u**3) / a1E
slist = [
s[0]
for s in (x**2 + a1E * x + (r + a2E + a2F * u**2)).roots()
]
for s in slist:
t = (a4E + a4F * u**4 + s * a3E + r * s * a1E + r**2)
if JustOne: return (u, r, s, t)
ans.append((u, r, s, t))
if JustOne: return None
ans.sort()
return ans
b2E, b4E, b6E, b8E = E.b_invariants()
b2F, b4F, b6F, b8F = F.b_invariants()
if char == 3:
if j == 0:
ulist = (x**4 - (b4E / b4F)).roots(multiplicities=False)
ans = []
for u in ulist:
s = a1E - a1F * u
t = a3E - a3F * u**3
rlist = (x**3 - b4E * x +
(b6E - b6F * u**6)).roots(multiplicities=False)
for r in rlist:
if JustOne: return (u, r, s, t + r * a1E)
ans.append((u, r, s, t + r * a1E))
if JustOne: return None
ans.sort()
return ans
else:
ulist = (x**2 - (b2E / b2F)).roots(multiplicities=False)
ans = []
for u in ulist:
r = (b4F * u**4 - b4E) / b2E
s = (a1E - a1F * u)
t = (a3E - a3F * u**3 + a1E * r)
if JustOne: return (u, r, s, t)
ans.append((u, r, s, t))
if JustOne: return None
ans.sort()
return ans
# now char!=2,3:
c4E, c6E = E.c_invariants()
c4F, c6F = F.c_invariants()
if j == 0:
m, um = 6, c6E / c6F
elif j == 1728:
m, um = 4, c4E / c4F
else:
m, um = 2, (c6E * c4F) / (c6F * c4E)
ulist = (x**m - um).roots(multiplicities=False)
ans = []
for u in ulist:
s = (a1F * u - a1E) / 2
r = (a2F * u**2 + a1E * s + s**2 - a2E) / 3
t = (a3F * u**3 - a1E * r - a3E) / 2
if JustOne: return (u, r, s, t)
ans.append((u, r, s, t))
if JustOne: return None
ans.sort()
return ans
def isomorphisms(E,F,JustOne=False):
r"""
Returns one or all isomorphisms between two elliptic curves.
INPUT:
- ``E``, ``F`` (EllipticCurve) -- Two elliptic curves.
- ``JustOne`` (bool) If True, returns one isomorphism, or None if
the curves are not isomorphic. If False, returns a (possibly
empty) list of isomorphisms.
OUTPUT:
Either None, or a 4-tuple `(u,r,s,t)` representing an isomorphism,
or a list of these.
.. note::
This function is not intended for users, who should use the
interface provided by ``ell_generic``.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.weierstrass_morphism import *
sage: isomorphisms(EllipticCurve_from_j(0),EllipticCurve('27a3'))
[(-1, 0, 0, -1), (1, 0, 0, 0)]
sage: isomorphisms(EllipticCurve_from_j(0),EllipticCurve('27a3'),JustOne=True)
(1, 0, 0, 0)
sage: isomorphisms(EllipticCurve_from_j(0),EllipticCurve('27a1'))
[]
sage: isomorphisms(EllipticCurve_from_j(0),EllipticCurve('27a1'),JustOne=True)
"""
from ell_generic import is_EllipticCurve
if not is_EllipticCurve(E) or not is_EllipticCurve(F):
raise ValueError("arguments are not elliptic curves")
K = E.base_ring()
# if not K == F.base_ring(): return []
j=E.j_invariant()
if j != F.j_invariant():
if JustOne: return None
return []
from sage.rings.all import PolynomialRing
x=PolynomialRing(K,'x').gen()
a1E, a2E, a3E, a4E, a6E = E.ainvs()
a1F, a2F, a3F, a4F, a6F = F.ainvs()
char=K.characteristic()
if char==2:
if j==0:
ulist=(x**3-(a3E/a3F)).roots(multiplicities=False)
ans=[]
for u in ulist:
slist=(x**4+a3E*x+(a2F**2+a4F)*u**4+a2E**2+a4E).roots(multiplicities=False)
for s in slist:
r=s**2+a2E+a2F*u**2
tlist= (x**2 + a3E*x + r**3 + a2E*r**2 + a4E*r + a6E + a6F*u**6).roots(multiplicities=False)
for t in tlist:
if JustOne: return (u,r,s,t)
ans.append((u,r,s,t))
if JustOne: return None
ans.sort()
return ans
else:
ans=[]
u=a1E/a1F
r=(a3E+a3F*u**3)/a1E
slist=[s[0] for s in (x**2+a1E*x+(r+a2E+a2F*u**2)).roots()]
for s in slist:
t = (a4E+a4F*u**4 + s*a3E + r*s*a1E + r**2)
if JustOne: return (u,r,s,t)
ans.append((u,r,s,t))
if JustOne: return None
ans.sort()
return ans
b2E, b4E, b6E, b8E = E.b_invariants()
b2F, b4F, b6F, b8F = F.b_invariants()
if char==3:
if j==0:
ulist=(x**4-(b4E/b4F)).roots(multiplicities=False)
ans=[]
for u in ulist:
s=a1E-a1F*u
t=a3E-a3F*u**3
rlist=(x**3-b4E*x+(b6E-b6F*u**6)).roots(multiplicities=False)
for r in rlist:
if JustOne: return (u,r,s,t+r*a1E)
ans.append((u,r,s,t+r*a1E))
if JustOne: return None
ans.sort()
return ans
else:
ulist=(x**2-(b2E/b2F)).roots(multiplicities=False)
ans=[]
for u in ulist:
r = (b4F*u**4 -b4E)/b2E
s = (a1E-a1F*u)
t = (a3E-a3F*u**3 + a1E*r)
if JustOne: return (u,r,s,t)
ans.append((u,r,s,t))
if JustOne: return None
ans.sort()
return ans
# now char!=2,3:
c4E,c6E = E.c_invariants()
c4F,c6F = F.c_invariants()
if j==0:
m,um = 6,c6E/c6F
elif j==1728:
m,um=4,c4E/c4F
else:
m,um=2,(c6E*c4F)/(c6F*c4E)
ulist=(x**m-um).roots(multiplicities=False)
ans=[]
for u in ulist:
s = (a1F*u - a1E)/2
r = (a2F*u**2 + a1E*s + s**2 - a2E)/3
t = (a3F*u**3 - a1E*r - a3E)/2
if JustOne: return (u,r,s,t)
ans.append((u,r,s,t))
if JustOne: return None
ans.sort()
return ans
