def from_weierstrass_map(self):
"""
Returns a morphism from the associated elliptic curve on
Weierstrass form to this Edwards curve.
"""
P2 = ProjectiveSpace(2, self.__base_ring, names='xyz')
x, y, z = P2.coordinate_ring().gens()
a, d = self.ainvs()
E = self.associated_ec()
C = P2.subscheme(a * x**2 * z**2 + y**2 * z**2 - z**4 -
d * x**2 * y**2)
f = WeierstrassTransformation(
E,
C,
[
2 * x * (x + (a - d) * z), # <-- x
(x - (a - d) * z) * y, # <-- y
y * (x + (a - d) * z) # <-- z
],
1)
return f
def to_weierstrass_map(self):
"""
Returns a morphism from this Edwards curve to the associated
elliptic curve on Weierstrass form.
"""
P2 = ProjectiveSpace(2, self.__base_ring, names='xyz')
x, y, z = P2.coordinate_ring().gens()
a, d = self.ainvs()
E = self.associated_ec()
C = P2.subscheme(a * x**2 * z**2 + y**2 * z**2 - z**4 -
d * x**2 * y**2)
f = WeierstrassTransformation(
C,
E,
[
(a - d) * (z + y) * x, # <-- x
(a - d) * 2 * (z**2 + y * z), # <-- y
z * x * (z - y) # <-- z
],
1)
return f
def Jacobian_of_equation(polynomial, variables=None, curve=None):
r"""
Construct the Jacobian of a genus-one curve given by a polynomial.
INPUT:
- ``F`` -- a polynomial defining a plane curve of genus one. May
be homogeneous or inhomogeneous.
- ``variables`` -- list of two or three variables or ``None``
(default). The inhomogeneous or homogeneous coordinates. By
default, all variables in the polynomial are used.
- ``curve`` -- the genus-one curve defined by ``polynomial`` or #
``None`` (default). If specified, suitable morphism from the
jacobian elliptic curve to the curve is returned.
OUTPUT:
An elliptic curve in short Weierstrass form isomorphic to the
curve ``polynomial=0``. If the optional argument ``curve`` is
specified, a rational multicover from the Jacobian elliptic curve
to the genus-one curve is returned.
EXAMPLES::
sage: R.<a,b,c> = QQ[]
sage: f = a^3+b^3+60*c^3
sage: Jacobian(f)
Elliptic Curve defined by y^2 = x^3 - 24300 over Rational Field
sage: Jacobian(f.subs(c=1))
Elliptic Curve defined by y^2 = x^3 - 24300 over Rational Field
If we specify the domain curve the birational covering is returned::
sage: h = Jacobian(f, curve=Curve(f)); h
Scheme morphism:
From: Projective Plane Curve over Rational Field defined by a^3 + b^3 + 60*c^3
To: Elliptic Curve defined by y^2 = x^3 - 24300 over Rational Field
Defn: Defined on coordinates by sending (a : b : c) to
(-216000*a^4*b^4*c - 12960000*a^4*b*c^4 - 12960000*a*b^4*c^4 :
108000*a^6*b^3 - 108000*a^3*b^6 - 6480000*a^6*c^3 + 6480000*b^6*c^3 + 388800000*a^3*c^6 - 388800000*b^3*c^6 :
216000*a^3*b^3*c^3)
sage: h([1,-1,0])
(0 : 1 : 0)
Plugging in the polynomials defining `h` allows us to verify that
it is indeed a rational morphism to the elliptic curve::
sage: E = h.codomain()
sage: E.defining_polynomial()(h.defining_polynomials()).factor()
(2519424000000000) * c^3 * b^3 * a^3 * (a^3 + b^3 + 60*c^3) *
(a^9*b^6 + a^6*b^9 - 120*a^9*b^3*c^3 + 900*a^6*b^6*c^3 - 120*a^3*b^9*c^3 +
3600*a^9*c^6 + 54000*a^6*b^3*c^6 + 54000*a^3*b^6*c^6 + 3600*b^9*c^6 +
216000*a^6*c^9 - 432000*a^3*b^3*c^9 + 216000*b^6*c^9)
By specifying the variables, we can also construct an elliptic
curve over a polynomial ring::
sage: R.<u,v,t> = QQ[]
sage: Jacobian(u^3+v^3+t, variables=[u,v])
Elliptic Curve defined by y^2 = x^3 + (-27/4*t^2) over
Multivariate Polynomial Ring in u, v, t over Rational Field
TESTS::
sage: from sage.schemes.elliptic_curves.jacobian import Jacobian_of_equation
sage: Jacobian_of_equation(f, variables=[a,b,c])
Elliptic Curve defined by y^2 = x^3 - 24300 over Rational Field
"""
from sage.schemes.toric.weierstrass import WeierstrassForm
f, g = WeierstrassForm(polynomial, variables=variables)
try:
K = polynomial.base_ring()
f = K(f)
g = K(g)
except (TypeError, ValueError):
pass
E = EllipticCurve([f, g])
if curve is None:
return E
X, Y, Z = WeierstrassForm(polynomial, variables=variables, transformation=True)
from sage.schemes.elliptic_curves.weierstrass_transform import WeierstrassTransformation
return WeierstrassTransformation(curve, E, [X*Z, Y, Z**3], 1)