Construct elliptic curves as Jacobians¶

An elliptic curve is a genus one curve with a designated point. The Jacobian of a genus-one curve can be defined as the set of line bundles on the curve, and is isomorphic to the original genus-one curve. It is also an elliptic curve with the trivial line bundle as designated point. The utility of this construction is that we can construct elliptic curves without having to specify which point we take as the origin.

EXAMPLES:

Sage

sage: R.<u,v,w> = QQ[]
sage: Jacobian(u^3 + v^3 + w^3)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
sage: Jacobian(u^4 + v^4 + w^2)
Elliptic Curve defined by y^2 = x^3 - 4*x over Rational Field

sage: C = Curve(u^3 + v^3 + w^3)
sage: Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

sage: P2.<u,v,w> = ProjectiveSpace(2, QQ)
sage: C = P2.subscheme(u^3 + v^3 + w^3)
sage: Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

Python

>>> from sage.all import *
>>> R = QQ['u, v, w']; (u, v, w,) = R._first_ngens(3)
>>> Jacobian(u**Integer(3) + v**Integer(3) + w**Integer(3))
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
>>> Jacobian(u**Integer(4) + v**Integer(4) + w**Integer(2))
Elliptic Curve defined by y^2 = x^3 - 4*x over Rational Field

>>> C = Curve(u**Integer(3) + v**Integer(3) + w**Integer(3))
>>> Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

>>> P2 = ProjectiveSpace(Integer(2), QQ, names=('u', 'v', 'w',)); (u, v, w,) = P2._first_ngens(3)
>>> C = P2.subscheme(u**Integer(3) + v**Integer(3) + w**Integer(3))
>>> Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

One can also define Jacobians of varieties that are not genus-one curves. These are not implemented in this module, but we call the relevant functionality:

Sage

sage: R.<x> = PolynomialRing(QQ)
sage: f = x**5 + 1184*x**3 + 1846*x**2 + 956*x + 560
sage: C = HyperellipticCurve(f)
sage: Jacobian(C)
Jacobian of Hyperelliptic Curve over Rational Field defined
 by y^2 = x^5 + 1184*x^3 + 1846*x^2 + 956*x + 560

Python

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> f = x**Integer(5) + Integer(1184)*x**Integer(3) + Integer(1846)*x**Integer(2) + Integer(956)*x + Integer(560)
>>> C = HyperellipticCurve(f)
>>> Jacobian(C)
Jacobian of Hyperelliptic Curve over Rational Field defined
 by y^2 = x^5 + 1184*x^3 + 1846*x^2 + 956*x + 560

REFERENCES:

Wikipedia article Jacobian_variety

sage.schemes.elliptic_curves.jacobian.Jacobian(X, **kwds)[source]¶

Return the Jacobian.

INPUT:

X – polynomial, algebraic variety, or anything else that has a Jacobian elliptic curve
kwds – optional keyword arguments

The input X can be one of the following:

A polynomial, see Jacobian_of_equation() for details.
A curve, see Jacobian_of_curve() for details.

EXAMPLES:

Sage

sage: R.<u,v,w> = QQ[]
sage: Jacobian(u^3 + v^3 + w^3)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

sage: C = Curve(u^3 + v^3 + w^3)
sage: Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

sage: P2.<u,v,w> = ProjectiveSpace(2, QQ)
sage: C = P2.subscheme(u^3 + v^3 + w^3)
sage: Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

sage: Jacobian(C, morphism=True)
Scheme morphism:
  From: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
  u^3 + v^3 + w^3
  To:   Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
  Defn: Defined on coordinates by sending (u : v : w) to
        (-u^4*v^4*w - u^4*v*w^4 - u*v^4*w^4 :
         1/2*u^6*v^3 - 1/2*u^3*v^6 - 1/2*u^6*w^3 + 1/2*v^6*w^3 + 1/2*u^3*w^6 - 1/2*v^3*w^6 :
         u^3*v^3*w^3)

Python

>>> from sage.all import *
>>> R = QQ['u, v, w']; (u, v, w,) = R._first_ngens(3)
>>> Jacobian(u**Integer(3) + v**Integer(3) + w**Integer(3))
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

>>> C = Curve(u**Integer(3) + v**Integer(3) + w**Integer(3))
>>> Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

>>> P2 = ProjectiveSpace(Integer(2), QQ, names=('u', 'v', 'w',)); (u, v, w,) = P2._first_ngens(3)
>>> C = P2.subscheme(u**Integer(3) + v**Integer(3) + w**Integer(3))
>>> Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

>>> Jacobian(C, morphism=True)
Scheme morphism:
  From: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
  u^3 + v^3 + w^3
  To:   Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
  Defn: Defined on coordinates by sending (u : v : w) to
        (-u^4*v^4*w - u^4*v*w^4 - u*v^4*w^4 :
         1/2*u^6*v^3 - 1/2*u^3*v^6 - 1/2*u^6*w^3 + 1/2*v^6*w^3 + 1/2*u^3*w^6 - 1/2*v^3*w^6 :
         u^3*v^3*w^3)

sage.schemes.elliptic_curves.jacobian.Jacobian_of_curve(curve, morphism=False)[source]¶

Return the Jacobian of a genus-one curve.

INPUT:

curve – a one-dimensional algebraic variety of genus one

OUTPUT: its Jacobian elliptic curve

EXAMPLES:

Sage

sage: R.<u,v,w> = QQ[]
sage: C = Curve(u^3 + v^3 + w^3)
sage: Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

Python

>>> from sage.all import *
>>> R = QQ['u, v, w']; (u, v, w,) = R._first_ngens(3)
>>> C = Curve(u**Integer(3) + v**Integer(3) + w**Integer(3))
>>> Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

sage.schemes.elliptic_curves.jacobian.Jacobian_of_equation(polynomial, variables=None, curve=None)[source]¶

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

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

Python

>>> from sage.all import *
>>> R = QQ['a, b, c']; (a, b, c,) = R._first_ngens(3)
>>> f = a**Integer(3) + b**Integer(3) + Integer(60)*c**Integer(3)
>>> Jacobian(f)
Elliptic Curve defined by y^2 = x^3 - 24300 over Rational Field
>>> Jacobian(f.subs(c=Integer(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

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)

Python

>>> from sage.all import *
>>> 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)

>>> h([Integer(1),-Integer(1),Integer(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

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)

Python

>>> from sage.all import *
>>> E = h.codomain()
>>> 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

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

Python

>>> from sage.all import *
>>> R = QQ['u, v, t']; (u, v, t,) = R._first_ngens(3)
>>> Jacobian(u**Integer(3) + v**Integer(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