Elliptic curve constructor

AUTHORS:

  • William Stein (2005): Initial version
  • John Cremona (2008-01): EllipticCurve(j) fixed for all cases
sage.schemes.elliptic_curves.constructor.EllipticCurve(x=None, y=None, j=None, minimal_twist=True)

There are several ways to construct an elliptic curve:

y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.

  • 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 a1=a2=a3=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.

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 ticket #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 ticket #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 ticket #11773:

sage: E = EllipticCurve()
Traceback (most recent call last):
...
TypeError: invalid input to EllipticCurve constructor
sage.schemes.elliptic_curves.constructor.EllipticCurve_from_c4c6(c4, c6)

Return an elliptic curve with given c_4 and c_6 invariants.

EXAMPLES:

sage: E = EllipticCurve_from_c4c6(17, -2005)
sage: E
Elliptic Curve defined by y^2  = x^3 - 17/48*x + 2005/864 over Rational Field
sage: E.c_invariants()
(17, -2005)
sage.schemes.elliptic_curves.constructor.EllipticCurve_from_cubic(F, P)

Construct an elliptic curve from a ternary cubic with a rational point.

INPUT:

  • F – a homogeneous cubic in three variables with rational coefficients (either as a polynomial ring element or as a string) defining a smooth plane cubic curve.
  • P – a 3-tuple (x,y,z) defining a projective point on the curve F=0.

OUTPUT:

(elliptic curve) An elliptic curve (in minimal Weierstrass form) isomorphic to the curve F=0.

Note

USES MAGMA - This function will not work on computers that do not have magma installed.

TO DO: implement this without using MAGMA.

For a more general version, see the function EllipticCurve_from_plane_curve().

EXAMPLES:

First we find that the Fermat cubic is isomorphic to the curve with Cremona label 27a1:

sage: E = EllipticCurve_from_cubic('x^3 + y^3 + z^3', [1,-1,0])  # optional - magma
sage: E         # optional - magma
Elliptic Curve defined by y^2 + y = x^3 - 7 over Rational Field
sage: E.cremona_label()     # optional - magma
'27a1'

Next we find the minimal model and conductor of the Jacobian of the Selmer curve.

sage: E = EllipticCurve_from_cubic('u^3 + v^3 + 60*w^3', [1,-1,0])   # optional - magma
sage: E                # optional - magma
Elliptic Curve defined by y^2  = x^3 - 24300 over Rational Field
sage: E.conductor()    # optional - magma
24300
sage.schemes.elliptic_curves.constructor.EllipticCurve_from_j(j, minimal_twist=True)

Return an elliptic curve with given j-invariant.

INPUT:

  • j – an element of some field.
  • minimal_twist (boolean, default True) – If True and j is in \QQ, the curve returned is a minimal twist, i.e. has minimal conductor. If j is not in \QQ this parameter is ignored.

OUTPUT:

(elliptic curve) An elliptic curve with j-invariant j.

EXAMPLES:

   sage: E = EllipticCurve_from_j(0); E; E.j_invariant(); E.label()
   Elliptic Curve defined by y^2 + y = x^3 over Rational Field
   0
   '27a3'

   sage: E = EllipticCurve_from_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_from_j(1); E; E.j_invariant()
   Elliptic Curve defined by y^2 + x*y = x^3 + 36*x + 3455 over Rational Field
   1

The ``minimal_twist`` parameter (ignored except over `\QQ` and
True by default) controls whether or not a minimal twist is
computed::

   sage: EllipticCurve_from_j(100)
   Elliptic Curve defined by y^2 = x^3 + x^2 + 3392*x + 307888 over Rational Field
   sage: _.conductor()
   33129800
   sage: EllipticCurve_from_j(100, minimal_twist=False)
   Elliptic Curve defined by y^2 = x^3 + 488400*x - 530076800 over Rational Field
   sage: _.conductor()
   298168200

Since computing the minimal twist requires factoring both j and j-1728 the following example would take a long time without setting minimal_twist to False:

sage: E = EllipticCurve_from_j(2^256+1,minimal_twist=False)
sage: E.j_invariant() == 2^256+1
True
sage.schemes.elliptic_curves.constructor.EllipticCurve_from_plane_curve(C, P)

Construct an elliptic curve from a smooth plane cubic with a rational point.

INPUT:

  • C – a plane curve of genus one.
  • P – a 3-tuple (x,y,z) defining a projective point on the curve C.

OUTPUT:

(elliptic curve) An elliptic curve (in minimal Weierstrass form) isomorphic to C.

Note

USES MAGMA - This function will not work on computers that do not have magma installed.

TO DO: implement this without using MAGMA.

EXAMPLES:

First we check that the Fermat cubic is isomorphic to the curve with Cremona label ‘27a1’:

sage: x,y,z=PolynomialRing(QQ,3,'xyz').gens() # optional - magma 
sage: C=Curve(x^3+y^3+z^3) # optional - magma
sage: P=C(1,-1,0) # optional - magma
sage: E=EllipticCurve_from_plane_curve(C,P) # optional - magma
sage: E # optional - magma
Elliptic Curve defined by y^2 + y = x^3 - 7 over Rational Field
sage: E.label() # optional - magma
'27a1'

Now we try a quartic example:

sage: u,v,w=PolynomialRing(QQ,3,'uvw').gens() # optional - magma 
sage: C=Curve(u^4+u^2*v^2-w^4) # optional - magma
sage: P=C(1,0,1) # optional - magma
sage: E=EllipticCurve_from_plane_curve(C,P) # optional - magma
sage: E # optional - magma
Elliptic Curve defined by y^2  = x^3 + 4*x over Rational Field
sage: E.label() # optional - magma
'32a1'
sage.schemes.elliptic_curves.constructor.EllipticCurves_with_good_reduction_outside_S(S=[], proof=None, verbose=False)

Returns a sorted list of all elliptic curves defined over Q with good reduction outside the set S of primes.

INPUT:

  • S - list of primes (default: empty list).
  • proof - True/False (default True): the MW basis for auxiliary curves will be computed with this proof flag.
  • verbose - True/False (default False): if True, some details of the computation will be output.

Note

Proof flag: The algorithm used requires determining all S-integral points on several auxiliary curves, which in turn requires the computation of their generators. This is not always possible (even in theory) using current knowledge.

The value of this flag is passed to the function which computes generators of various auxiliary elliptic curves, in order to find their S-integral points. Set to False if the default (True) causes warning messages, but note that you can then not rely on the set of curves returned being complete.

EXAMPLES:

sage: EllipticCurves_with_good_reduction_outside_S([])
[]
sage: elist = EllipticCurves_with_good_reduction_outside_S([2])
sage: elist
[Elliptic Curve defined by y^2 = x^3 + 4*x over Rational Field,
Elliptic Curve defined by y^2 = x^3 - x over Rational Field,
...
Elliptic Curve defined by y^2 = x^3 - x^2 - 13*x + 21 over Rational Field]
sage: len(elist)
24
sage: ', '.join([e.label() for e in elist])
'32a1, 32a2, 32a3, 32a4, 64a1, 64a2, 64a3, 64a4, 128a1, 128a2, 128b1, 128b2, 128c1, 128c2, 128d1, 128d2, 256a1, 256a2, 256b1, 256b2, 256c1, 256c2, 256d1, 256d2'

Without Proof=False, this example gives two warnings:

sage: elist = EllipticCurves_with_good_reduction_outside_S([11],proof=False)  # long time (14s on sage.math, 2011)
sage: len(elist)  # long time
12
sage: ', '.join([e.label() for e in elist])  # long time
'11a1, 11a2, 11a3, 121a1, 121a2, 121b1, 121b2, 121c1, 121c2, 121d1, 121d2, 121d3'

sage: elist = EllipticCurves_with_good_reduction_outside_S([2,3]) # long time (26s on sage.math, 2011)
sage: len(elist) # long time
752
sage: max([e.conductor() for e in elist]) # long time
62208
sage: [N.factor() for N in Set([e.conductor() for e in elist])] # long time
[2^7,
2^8,
2^3 * 3^4,
2^2 * 3^3,
2^8 * 3^4,
2^4 * 3^4,
2^3 * 3,
2^7 * 3,
2^3 * 3^5,
3^3,
2^8 * 3,
2^5 * 3^4,
2^4 * 3,
2 * 3^4,
2^2 * 3^2,
2^6 * 3^4,
2^6,
2^7 * 3^2,
2^4 * 3^5,
2^4 * 3^3,
2 * 3^3,
2^6 * 3^3,
2^6 * 3,
2^5,
2^2 * 3^4,
2^3 * 3^2,
2^5 * 3,
2^7 * 3^4,
2^2 * 3^5,
2^8 * 3^2,
2^5 * 3^2,
2^7 * 3^5,
2^8 * 3^5,
2^3 * 3^3,
2^8 * 3^3,
2^5 * 3^5,
2^4 * 3^2,
2 * 3^5,
2^5 * 3^3,
2^6 * 3^5,
2^7 * 3^3,
3^5,
2^6 * 3^2]

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