def descend_to(self, K, f=None):
r"""
Given a subfield `K` and an elliptic curve self defined over a field `L`,
this function determines whether there exists an elliptic curve over `K`
which is isomorphic over `L` to self. If one exists, it finds it.
INPUT:
- `K` -- a subfield of the base field of self.
- `f` -- an embedding of `K` into the base field of self.
OUTPUT:
Either an elliptic curve defined over `K` which is isomorphic to self
or None if no such curve exists.
.. NOTE::
This only works over number fields and QQ.
EXAMPLES::
sage: E = EllipticCurve([1,2,3,4,5])
sage: E.descend_to(ZZ)
Traceback (most recent call last):
...
TypeError: Input must be a field.
::
sage: F.<b> = QuadraticField(23)
sage: G.<a> = F.extension(x^3+5)
sage: E = EllipticCurve(j=1728*b).change_ring(G)
sage: E.descend_to(F)
Elliptic Curve defined by y^2 = x^3 + (8957952*b-206032896)*x + (-247669456896*b+474699792384) over Number Field in b with defining polynomial x^2 - 23
::
sage: L.<a> = NumberField(x^4 - 7)
sage: K.<b> = NumberField(x^2 - 7)
sage: E = EllipticCurve([a^6,0])
sage: E.descend_to(K)
Elliptic Curve defined by y^2 = x^3 + 1296/49*b*x over Number Field in b with defining polynomial x^2 - 7
::
sage: K.<a> = QuadraticField(17)
sage: E = EllipticCurve(j = 2*a)
sage: print E.descend_to(QQ)
None
"""
if not K.is_field():
raise TypeError, "Input must be a field."
if self.base_field()==K:
return self
j = self.j_invariant()
from sage.rings.all import QQ
if K == QQ:
f = QQ.embeddings(self.base_field())[0]
if j in QQ:
jbase = QQ(j)
else:
return None
elif f == None:
embeddings = K.embeddings(self.base_field())
if len(embeddings) == 0:
raise TypeError, "Input must be a subfield of the base field of the curve."
for g in embeddings:
try:
jbase = g.preimage(j)
f = g
break
except StandardError:
pass
if f == None:
return None
else:
try:
jbase = f.preimage(j)
except StandardError:
return None
E = EllipticCurve(j=jbase)
E2 = EllipticCurve(self.base_field(), [f(a) for a in E.a_invariants()])
if jbase==0:
d = self.is_sextic_twist(E2)
if d == 1:
return E
if d == 0:
return None
Etwist = E2.sextic_twist(d)
elif jbase==1728:
d = self.is_quartic_twist(E2)
if d == 1:
return E
if d == 0:
return None
Etwist = E2.quartic_twist(d)
else:
d = self.is_quadratic_twist(E2)
if d == 1:
return E
if d == 0:
return None
Etwist = E2.quadratic_twist(d)
if Etwist.is_isomorphic(self):
try:
Eout = EllipticCurve(K, [f.preimage(a) for a in Etwist.a_invariants()])
except StandardError:
return None
else:
return Eout
def descend_to(self, K, f=None):
r"""
Given a subfield `K` and an elliptic curve self defined over a field `L`,
this function determines whether there exists an elliptic curve over `K`
which is isomorphic over `L` to self. If one exists, it finds it.
INPUT:
- `K` -- a subfield of the base field of self.
- `f` -- an embedding of `K` into the base field of self.
OUTPUT:
Either an elliptic curve defined over `K` which is isomorphic to self
or None if no such curve exists.
.. NOTE::
This only works over number fields and QQ.
EXAMPLES::
sage: E = EllipticCurve([1,2,3,4,5])
sage: E.descend_to(ZZ)
Traceback (most recent call last):
...
TypeError: Input must be a field.
::
sage: F.<b> = QuadraticField(23)
sage: G.<a> = F.extension(x^3+5)
sage: E = EllipticCurve(j=1728*b).change_ring(G)
sage: E.descend_to(F)
Elliptic Curve defined by y^2 = x^3 + (8957952*b-206032896)*x + (-247669456896*b+474699792384) over Number Field in b with defining polynomial x^2 - 23
::
sage: L.<a> = NumberField(x^4 - 7)
sage: K.<b> = NumberField(x^2 - 7)
sage: E = EllipticCurve([a^6,0])
sage: E.descend_to(K)
Elliptic Curve defined by y^2 = x^3 + 1296/49*b*x over Number Field in b with defining polynomial x^2 - 7
::
sage: K.<a> = QuadraticField(17)
sage: E = EllipticCurve(j = 2*a)
sage: print E.descend_to(QQ)
None
"""
if not K.is_field():
raise TypeError, "Input must be a field."
if self.base_field() == K:
return self
j = self.j_invariant()
from sage.rings.all import QQ
if K == QQ:
f = QQ.embeddings(self.base_field())[0]
if j in QQ:
jbase = QQ(j)
else:
return None
elif f == None:
embeddings = K.embeddings(self.base_field())
if len(embeddings) == 0:
raise TypeError, "Input must be a subfield of the base field of the curve."
for g in embeddings:
try:
jbase = g.preimage(j)
f = g
break
except StandardError:
pass
if f == None:
return None
else:
try:
jbase = f.preimage(j)
except StandardError:
return None
E = EllipticCurve(j=jbase)
E2 = EllipticCurve(self.base_field(), [f(a) for a in E.a_invariants()])
if jbase == 0:
d = self.is_sextic_twist(E2)
if d == 1:
return E
if d == 0:
return None
Etwist = E2.sextic_twist(d)
elif jbase == 1728:
d = self.is_quartic_twist(E2)
if d == 1:
return E
if d == 0:
return None
Etwist = E2.quartic_twist(d)
else:
d = self.is_quadratic_twist(E2)
if d == 1:
return E
if d == 0:
return None
Etwist = E2.quadratic_twist(d)
if Etwist.is_isomorphic(self):
try:
Eout = EllipticCurve(
K, [f.preimage(a) for a in Etwist.a_invariants()])
except StandardError:
return None
else:
return Eout
