AUTHORS:
There are several ways to construct an elliptic curve:
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 with
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 with
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 -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 is given, the constructed
curve is a minimal twist (minimal conductor for curves with that
-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 and
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
Return an elliptic curve with given and
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)
Construct an elliptic curve from a ternary cubic with a rational point.
INPUT:
OUTPUT:
(elliptic curve) An elliptic curve (in minimal Weierstrass form)
isomorphic to the curve .
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
Return an elliptic curve with given -invariant.
INPUT:
OUTPUT:
(elliptic curve) An elliptic curve with -invariant
.
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 and
the following example would take a long time without
setting
to False:
sage: E = EllipticCurve_from_j(2^256+1,minimal_twist=False)
sage: E.j_invariant() == 2^256+1
True
Construct an elliptic curve from a smooth plane cubic with a rational point.
INPUT:
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'
Returns a sorted list of all elliptic curves defined over
with good reduction outside the set
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]