Elliptic curve constructor

AUTHORS:

• William Stein (2005): Initial version
• John Cremona (2008-01): EllipticCurve(j) fixed for all cases

class sage.schemes.elliptic_curves.constructor.EllipticCurveFactory

Bases: sage.structure.factory.UniqueFactory

Construct an elliptic curve.

In Sage, an elliptic curve is always specified by (the coefficients of) a long Weierstrass equation

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

INPUT:

There are several ways to construct an elliptic curve:

• EllipticCurve([a1,a2,a3,a4,a6]): Elliptic curve with given \(a\)-invariants. The invariants are coerced into a common parent. If all are integers, they are coerced into the rational numbers.
• EllipticCurve([a4,a6]): Same as above, but \(a_1=a_2=a_3=0\).
• EllipticCurve(label): Returns the elliptic curve over \(\QQ\) 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 commutative ring, although most functionality is only implemented over fields.
• EllipticCurve(j=j0) or EllipticCurve_from_j(j0): Return an elliptic curve with \(j\)-invariant j0.
• EllipticCurve(polynomial): Read off the \(a\)-invariants from the polynomial coefficients, see EllipticCurve_from_Weierstrass_polynomial().

Instead of giving the coefficients as a list of length 2 or 5, one can also give a tuple.

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 ticket #6657

sage: EllipticCurve(GF(144169),j=1728)
Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 144169

Elliptic curves over the same ring with the same Weierstrass coefficients are identical, even when they are constructed in different ways (see trac ticket #11474):

sage: EllipticCurve('11a3') is EllipticCurve(QQ, [0, -1, 1, 0, 0])
True

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

create_key_and_extra_args(x=None, y=None, j=None, minimal_twist=True, **kwds)

Return a UniqueFactory key and possibly extra parameters.

INPUT:

See the documentation for EllipticCurveFactory.

OUTPUT:

A pair (key, extra_args):

• key has the form \((R, (a_1, a_2, a_3, a_4, a_6))\), representing a ring and the Weierstrass coefficients of an elliptic curve over that ring;
• extra_args is a dictionary containing additional data to be inserted into the elliptic curve structure.

EXAMPLES:

sage: EllipticCurve.create_key_and_extra_args(j=8000)
((Rational Field, (0, -1, 0, -3, -1)), {})

When constructing a curve over \(\QQ\) from a Cremona or LMFDB label, the invariants from the database are returned as extra_args:

sage: key, data = EllipticCurve.create_key_and_extra_args('389.a1')
sage: key
(Rational Field, (0, 1, 1, -2, 0))
sage: data['conductor']
389
sage: data['cremona_label']
'389a1'
sage: data['lmfdb_label']
'389.a1'
sage: data['rank']
2
sage: data['torsion_order']
1

User-specified keywords are also included in extra_args:

sage: key, data = EllipticCurve.create_key_and_extra_args((0, 0, 1, -23737, 960366), rank=4)
sage: data['rank']
4

Furthermore, keywords takes precedence over data from the database, which can be used to specify an alternative set of generators for the Mordell-Weil group:

sage: key, data = EllipticCurve.create_key_and_extra_args('5077a1', gens=[[1, -1], [-2, 3], [4, -7]])
sage: data['gens']
[[1, -1], [-2, 3], [4, -7]]
sage: E = EllipticCurve.create_object(0, key, **data)
sage: E.gens()
[(-2 : 3 : 1), (1 : -1 : 1), (4 : -7 : 1)]

Note that elliptic curves are equal if and only they have the same base ring and Weierstrass equation; the data in extra_args do not influence comparison of elliptic curves. A consequence of this is that passing keyword arguments only works when constructing an elliptic curve the first time:

sage: E = EllipticCurve('433a1', gens=[[-1, 1], [3, 4]])
sage: E.gens()
[(-1 : 1 : 1), (3 : 4 : 1)]
sage: E = EllipticCurve('433a1', gens=[[-1, 0], [0, 1]])
sage: E.gens()
[(-1 : 1 : 1), (3 : 4 : 1)]

Warning

Manually specifying extra data is almost never necessary and is not guaranteed to have any effect, as the above example shows. Almost no checking is done, so specifying incorrect data may lead to wrong results of computations instead of errors or warnings.

create_object(version, key, **kwds)

Create an object from a UniqueFactory key.

EXAMPLES:

sage: E = EllipticCurve.create_object(0, (GF(3), (1, 2, 0, 1, 2)))
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_finite_field.EllipticCurve_finite_field_with_category'>

Note

Keyword arguments are currently only passed to the constructor for elliptic curves over \(\QQ\); elliptic curves over other fields do not support them.

sage.schemes.elliptic_curves.constructor.EllipticCurve_from_Weierstrass_polynomial(f)

Return the elliptic curve defined by a cubic in (long) Weierstrass form.

INPUT:

• f – a inhomogeneous cubic polynomial in long Weierstrass form.

OUTPUT:

The elliptic curve defined by it.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: f = y^2 + 1*x*y + 3*y - (x^3 + 2*x^2 + 4*x + 6)
sage: EllipticCurve(f)
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 6 over Rational Field
sage: EllipticCurve(f).a_invariants()
(1, 2, 3, 4, 6)

The polynomial ring may have extra variables as long as they do not occur in the polynomial itself:

sage: R.<x,y,z,w> = QQ[]
sage: EllipticCurve(-y^2 + x^3 + 1)
Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field
sage: EllipticCurve(-x^2 + y^3 + 1)
Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field
sage: EllipticCurve(-w^2 + z^3 + 1)
Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field

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, morphism=True)

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

If you just want the Weierstrass form and are not interested in the morphism then it is easier to use Jacobian() instead. This will construct the same elliptic curve but you don’t have to supply the point P.

INPUT:

• F – a homogeneous cubic in three variables with rational coefficients, as a polynomial ring element, defining a smooth plane cubic curve.
• P – a 3-tuple \((x,y,z)\) defining a projective point on the curve \(F=0\). Need not be a flex, but see caveat on output.
• morphism – boolean (default: True). Whether to return the morphism or just the elliptic curve.

OUTPUT:

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

If morphism=True is passed, then a birational equivalence between F and the Weierstrass curve is returned. If the point happens to be a flex, then this is an isomorphism.

EXAMPLES:

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

sage: R.<x,y,z> = QQ[]
sage: cubic = x^3+y^3+z^3
sage: P = [1,-1,0]
sage: E = EllipticCurve_from_cubic(cubic, P, morphism=False); E
Elliptic Curve defined by y^2 + 2*x*y + 1/3*y = x^3 - x^2 - 1/3*x - 1/27 over Rational Field
sage: E.cremona_label()
'27a1'
sage: EllipticCurve_from_cubic(cubic, [0,1,-1], morphism=False).cremona_label()
'27a1'
sage: EllipticCurve_from_cubic(cubic, [1,0,-1], morphism=False).cremona_label()
'27a1'

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

sage: R.<a,b,c> = QQ[]
sage: cubic = a^3+b^3+60*c^3
sage: P = [1,-1,0]
sage: E = EllipticCurve_from_cubic(cubic, P, morphism=False);  E
Elliptic Curve defined by y^2 + 2*x*y + 20*y = x^3 - x^2 - 20*x - 400/3 over Rational Field
sage: E.minimal_model()
Elliptic Curve defined by y^2 = x^3 - 24300 over Rational Field
sage: E.conductor()
24300

We can also get the birational equivalence to and from the Weierstrass form. We start with an example where P is a flex and the equivalence is an isomorphism:

sage: f = EllipticCurve_from_cubic(cubic, P, morphism=True)
sage: f
Scheme morphism:
  From: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
        a^3 + b^3 + 60*c^3
  To:   Elliptic Curve defined by y^2 + 2*x*y + 20*y = x^3 - x^2 - 20*x - 400/3
        over Rational Field
  Defn: Defined on coordinates by sending (a : b : c) to
        (-c : -b + c : 1/20*a + 1/20*b)

sage: finv = f.inverse();  finv
Scheme morphism:
  From: Elliptic Curve defined by y^2 + 2*x*y + 20*y = x^3 - x^2 - 20*x - 400/3
        over Rational Field
  To:   Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
  a^3 + b^3 + 60*c^3
  Defn: Defined on coordinates by sending (x : y : z) to
        (x + y + 20*z : -x - y : -x)

We verify that \(f\) maps the chosen point \(P=(1,-1,0)\) on the cubic to the origin of the elliptic curve:

sage: f([1,-1,0])
(0 : 1 : 0)
sage: finv([0,1,0])
(-1 : 1 : 0)

To verify the output, we plug in the polynomials to check that this indeed transforms the cubic into Weierstrass form:

sage: cubic(finv.defining_polynomials()) * finv.post_rescaling()
-x^3 + x^2*z + 2*x*y*z + y^2*z + 20*x*z^2 + 20*y*z^2 + 400/3*z^3

sage: E.defining_polynomial()(f.defining_polynomials()) * f.post_rescaling()
a^3 + b^3 + 60*c^3

If the point is not a flex then the cubic can not be transformed to a Weierstrass equation by a linear transformation. The general birational transformation is quadratic:

sage: cubic =  a^3+7*b^3+64*c^3
sage: P = [2,2,-1]
sage: f = EllipticCurve_from_cubic(cubic, P, morphism=True)
sage: E = f.codomain();  E
Elliptic Curve defined by y^2 - 722*x*y - 21870000*y = x^3
+ 23579*x^2 over Rational Field
sage: E.minimal_model()
Elliptic Curve defined by y^2 + y = x^3 - 331 over Rational Field

sage: f
Scheme morphism:
  From: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
        a^3 + 7*b^3 + 64*c^3
  To:   Elliptic Curve defined by y^2 - 722*x*y - 21870000*y =
        x^3 + 23579*x^2 over Rational Field
  Defn: Defined on coordinates by sending (a : b : c) to
        (-5/112896*a^2 - 17/40320*a*b - 1/1280*b^2 - 29/35280*a*c
         - 13/5040*b*c - 4/2205*c^2 :
         -4055/112896*a^2 - 4787/40320*a*b - 91/1280*b^2 - 7769/35280*a*c
         - 1993/5040*b*c - 724/2205*c^2 :
         1/4572288000*a^2 + 1/326592000*a*b + 1/93312000*b^2 + 1/142884000*a*c
         + 1/20412000*b*c + 1/17860500*c^2)

sage: finv = f.inverse();  finv
Scheme morphism:
  From: Elliptic Curve defined by y^2 - 722*x*y - 21870000*y =
        x^3 + 23579*x^2 over Rational Field
  To:   Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
        a^3 + 7*b^3 + 64*c^3
  Defn: Defined on coordinates by sending (x : y : z) to
        (2*x^2 + 227700*x*z - 900*y*z :
         2*x^2 - 32940*x*z + 540*y*z :
         -x^2 - 56520*x*z - 180*y*z)

sage: cubic(finv.defining_polynomials()) * finv.post_rescaling()
-x^3 - 23579*x^2*z - 722*x*y*z + y^2*z - 21870000*y*z^2

sage: E.defining_polynomial()(f.defining_polynomials()) * f.post_rescaling()
a^3 + 7*b^3 + 64*c^3

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:

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)

Deprecated way to construct an elliptic curve.

Use Jacobian() instead.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: C = Curve(x^3+y^3+z^3)
sage: P = C(1,-1,0)
sage: E = EllipticCurve_from_plane_curve(C,P); E  # long time (3s on sage.math, 2013)
doctest:...: DeprecationWarning: use Jacobian(C) instead
See http://trac.sagemath.org/3416 for details.
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field

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]

sage.schemes.elliptic_curves.constructor.are_projectively_equivalent(P, Q, base_ring)

Test whether P and Q are projectively equivalent.

INPUT:

• P, Q – list/tuple of projective coordinates.
• base_ring – the base ring.

OUTPUT:

Boolean.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.constructor import are_projectively_equivalent
sage: are_projectively_equivalent([0,1,2,3], [0,1,2,2], base_ring=QQ)
False
sage: are_projectively_equivalent([0,1,2,3], [0,2,4,6], base_ring=QQ)
True

sage.schemes.elliptic_curves.constructor.chord_and_tangent(F, P)

Use the chord and tangent method to get another point on a cubic.

INPUT:

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

OUTPUT:

Another point satisfying the equation F.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: from sage.schemes.elliptic_curves.constructor import chord_and_tangent
sage: F = x^3+y^3+60*z^3
sage: chord_and_tangent(F, [1,-1,0])
[1, -1, 0]

sage: F = x^3+7*y^3+64*z^3
sage: p0 = [2,2,-1]
sage: p1 = chord_and_tangent(F, p0);  p1
[-5, 3, -1]
sage: p2 = chord_and_tangent(F, p1);  p2
[1265, -183, -314]

sage.schemes.elliptic_curves.constructor.coefficients_from_Weierstrass_polynomial(f)

Return the coefficients \((a_1, a_2, a_3, a_4, a_5)\) for a cubic in Weierstrass form.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.constructor import coefficients_from_Weierstrass_polynomial
sage: R.<w,z> = QQ[]
sage: coefficients_from_Weierstrass_polynomial(-w^2 + z^3 + 1)
[0, 0, 0, 0, 1]

sage.schemes.elliptic_curves.constructor.coefficients_from_j(j, minimal_twist=True)

Return Weierstrass coefficients \((a_1, a_2, a_3, a_4, a_6)\) for an elliptic curve with given \(j\)-invariant.

INPUT:

See EllipticCurve_from_j().

EXAMPLES:

sage: from sage.schemes.elliptic_curves.constructor import coefficients_from_j
sage: coefficients_from_j(0)
[0, 0, 1, 0, 0]
sage: coefficients_from_j(1728)
[0, 0, 0, -1, 0]
sage: coefficients_from_j(1)
[1, 0, 0, 36, 3455]

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

sage: coefficients_from_j(100)
[0, 1, 0, 3392, 307888]
sage: coefficients_from_j(100, minimal_twist=False)
[0, 0, 0, 488400, -530076800]

sage.schemes.elliptic_curves.constructor.projective_point(p)

Return equivalent point with denominators removed

INPUT:

• P, Q – list/tuple of projective coordinates.

OUTPUT:

List of projective coordinates.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.constructor import projective_point
sage: projective_point([4/5, 6/5, 8/5])
[2, 3, 4]
sage: F = GF(11)
sage: projective_point([F(4), F(8), F(2)])
[4, 8, 2]