Elliptic curves over a general ring#
Sage defines an elliptic curve over a ring \(R\) as a Weierstrass Model with five coefficients \([a_1,a_2,a_3,a_4,a_6]\) in \(R\) given by
Note that the (usual) scheme-theoretic definition of an elliptic curve over \(R\) would require the discriminant to be a unit in \(R\); Sage only imposes that the discriminant is non-zero. Also note that in Magma, “Weierstrass Model” refers to a model with \(a_1=a_2=a_3=0\), which is called Short Weierstrass Model in Sage; these do not always exist in characteristics 2 and 3.
EXAMPLES:
We construct an elliptic curve over an elaborate base ring:
sage: p, a, b = 97, 1, 3
sage: R.<u> = GF(p)[]
sage: S.<v> = R[]
sage: T = S.fraction_field()
sage: E = EllipticCurve(T, [a, b]); E
Elliptic Curve defined by y^2 = x^3 + x + 3 over Fraction Field of Univariate Polynomial Ring in v over Univariate Polynomial Ring in u over Finite Field of size 97
sage: latex(E)
y^2 = x^{3} + x + 3
AUTHORS:
William Stein (2005): Initial version
Robert Bradshaw et al….
John Cremona (2008-01): isomorphisms, automorphisms and twists in all characteristics
Julian Rueth (2014-04-11): improved caching
Lorenz Panny (2022-04-14): added
.montgomery_model()
- class sage.schemes.elliptic_curves.ell_generic.EllipticCurve_generic(K, ainvs)#
Bases:
sage.misc.fast_methods.WithEqualityById
,sage.schemes.curves.projective_curve.ProjectivePlaneCurve
Elliptic curve over a generic base ring.
EXAMPLES:
sage: E = EllipticCurve([1,2,3/4,7,19]); E Elliptic Curve defined by y^2 + x*y + 3/4*y = x^3 + 2*x^2 + 7*x + 19 over Rational Field sage: loads(E.dumps()) == E True sage: E = EllipticCurve([1,3]) sage: P = E([-1,1,1]) sage: -5*P (179051/80089 : -91814227/22665187 : 1)
- a1()#
Return the \(a_1\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,6]) sage: E.a1() 1
- a2()#
Return the \(a_2\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,6]) sage: E.a2() 2
- a3()#
Return the \(a_3\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,6]) sage: E.a3() 3
- a4()#
Return the \(a_4\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,6]) sage: E.a4() 4
- a6()#
Return the \(a_6\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,6]) sage: E.a6() 6
- a_invariants()#
The \(a\)-invariants of this elliptic curve, as a tuple.
OUTPUT:
(tuple) - a 5-tuple of the \(a\)-invariants of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,5]) sage: E.a_invariants() (1, 2, 3, 4, 5) sage: E = EllipticCurve([0,1]); E Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field sage: E.a_invariants() (0, 0, 0, 0, 1) sage: E = EllipticCurve([GF(7)(3),5]) sage: E.a_invariants() (0, 0, 0, 3, 5)
- ainvs()#
The \(a\)-invariants of this elliptic curve, as a tuple.
OUTPUT:
(tuple) - a 5-tuple of the \(a\)-invariants of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,5]) sage: E.a_invariants() (1, 2, 3, 4, 5) sage: E = EllipticCurve([0,1]); E Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field sage: E.a_invariants() (0, 0, 0, 0, 1) sage: E = EllipticCurve([GF(7)(3),5]) sage: E.a_invariants() (0, 0, 0, 3, 5)
- automorphisms(field=None)#
Return the set of isomorphisms from self to itself (as a list).
INPUT:
field
(defaultNone
) – a field into which the coefficients of the curve may be coerced (by default, uses the base field of the curve).
OUTPUT:
(list) A list of
WeierstrassIsomorphism
objects consisting of all the isomorphisms from the curveself
to itself defined overfield
.EXAMPLES:
sage: E = EllipticCurve_from_j(QQ(0)) # a curve with j=0 over QQ sage: E.automorphisms(); [Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Rational Field Via: (u,r,s,t) = (-1, 0, 0, -1), Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Rational Field Via: (u,r,s,t) = (1, 0, 0, 0)]
We can also find automorphisms defined over extension fields:
sage: K.<a> = NumberField(x^2+3) # adjoin roots of unity sage: E.automorphisms(K) [Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3 Via: (u,r,s,t) = (-1, 0, 0, -1), ... Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3 Via: (u,r,s,t) = (1, 0, 0, 0)]
sage: [ len(EllipticCurve_from_j(GF(q,'a')(0)).automorphisms()) for q in [2,4,3,9,5,25,7,49]] [2, 24, 2, 12, 2, 6, 6, 6]
- b2()#
Return the \(b_2\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,5]) sage: E.b2() 9
- b4()#
Return the \(b_4\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,5]) sage: E.b4() 11
- b6()#
Return the \(b_6\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,5]) sage: E.b6() 29
- b8()#
Return the \(b_8\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,5]) sage: E.b8() 35
- b_invariants()#
Return the \(b\)-invariants of this elliptic curve, as a tuple.
OUTPUT:
(tuple) - a 4-tuple of the \(b\)-invariants of this elliptic curve.
This method is cached.
EXAMPLES:
sage: E = EllipticCurve([0, -1, 1, -10, -20]) sage: E.b_invariants() (-4, -20, -79, -21) sage: E = EllipticCurve([-4,0]) sage: E.b_invariants() (0, -8, 0, -16) sage: E = EllipticCurve([1,2,3,4,5]) sage: E.b_invariants() (9, 11, 29, 35) sage: E.b2() 9 sage: E.b4() 11 sage: E.b6() 29 sage: E.b8() 35
ALGORITHM:
These are simple functions of the \(a\)-invariants.
AUTHORS:
William Stein (2005-04-25)
- base_extend(R)#
Return the base extension of
self
to \(R\).INPUT:
R
– either a ring into which the \(a\)-invariants ofself
may be converted, or a morphism which may be applied to them.
OUTPUT:
An elliptic curve over the new ring whose \(a\)-invariants are the images of the \(a\)-invariants of
self
.EXAMPLES:
sage: E = EllipticCurve(GF(5),[1,1]); E Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 5 sage: E1 = E.base_extend(GF(125,'a')); E1 Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field in a of size 5^3
- base_ring()#
Return the base ring of the elliptic curve.
EXAMPLES:
sage: E = EllipticCurve(GF(49, 'a'), [3,5]) sage: E.base_ring() Finite Field in a of size 7^2
sage: E = EllipticCurve([1,1]) sage: E.base_ring() Rational Field
sage: E = EllipticCurve(ZZ, [3,5]) sage: E.base_ring() Integer Ring
- c4()#
Return the \(c_4\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([0, -1, 1, -10, -20]) sage: E.c4() 496
- c6()#
Return the \(c_6\) invariant of this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([0, -1, 1, -10, -20]) sage: E.c6() 20008
- c_invariants()#
Return the \(c\)-invariants of this elliptic curve, as a tuple.
This method is cached.
OUTPUT:
(tuple) - a 2-tuple of the \(c\)-invariants of the elliptic curve.
EXAMPLES:
sage: E = EllipticCurve([0, -1, 1, -10, -20]) sage: E.c_invariants() (496, 20008) sage: E = EllipticCurve([-4,0]) sage: E.c_invariants() (192, 0)
ALGORITHM:
These are simple functions of the \(a\)-invariants.
AUTHORS:
William Stein (2005-04-25)
- change_ring(R)#
Return the base change of
self
to \(R\).This has the same effect as
self.base_extend(R)
.EXAMPLES:
sage: F2 = GF(5^2,'a'); a = F2.gen() sage: F4 = GF(5^4,'b'); b = F4.gen() sage: h = F2.hom([a.charpoly().roots(ring=F4,multiplicities=False)[0]],F4) sage: E = EllipticCurve(F2,[1,a]); E Elliptic Curve defined by y^2 = x^3 + x + a over Finite Field in a of size 5^2 sage: E.change_ring(h) Elliptic Curve defined by y^2 = x^3 + x + (4*b^3+4*b^2+4*b+3) over Finite Field in b of size 5^4
- change_weierstrass_model(*urst)#
Return a new Weierstrass model of self under the standard transformation \((u,r,s,t)\)
\[(x,y) \mapsto (x',y') = (u^2x + r , u^3y + su^2x + t).\]EXAMPLES:
sage: E = EllipticCurve('15a') sage: F1 = E.change_weierstrass_model([1/2,0,0,0]); F1 Elliptic Curve defined by y^2 + 2*x*y + 8*y = x^3 + 4*x^2 - 160*x - 640 over Rational Field sage: F2 = E.change_weierstrass_model([7,2,1/3,5]); F2 Elliptic Curve defined by y^2 + 5/21*x*y + 13/343*y = x^3 + 59/441*x^2 - 10/7203*x - 58/117649 over Rational Field sage: F1.is_isomorphic(F2) True
- discriminant()#
Return the discriminant of this elliptic curve.
This method is cached.
EXAMPLES:
sage: E = EllipticCurve([0,0,1,-1,0]) sage: E.discriminant() 37 sage: E = EllipticCurve([0, -1, 1, -10, -20]) sage: E.discriminant() -161051 sage: E = EllipticCurve([GF(7)(2),1]) sage: E.discriminant() 1
- division_polynomial(m, x=None, two_torsion_multiplicity=2, force_evaluate=None)#
Return the \(m^{th}\) division polynomial of this elliptic curve evaluated at \(x\).
The division polynomial is cached if \(x\) is
None
.INPUT:
m
– positive integer.x
– optional ring element to use as the \(x\) variable. If \(x\) isNone
(omitted), then a new polynomial ring will be constructed over the base ring of the elliptic curve, and its generator will be used as \(x\). Note that \(x\) does not need to be a generator of a polynomial ring; any ring element works. This permits fast calculation of the torsion polynomial evaluated on any element of a ring.two_torsion_multiplicity
– 0, 1, or 2If 0: For even \(m\) when \(x\) is
None
, a univariate polynomial over the base ring of the curve is returned, which omits factors whose roots are the \(x\)-coordinates of the \(2\)-torsion points. When \(x\) is notNone
, the evaluation of such a polynomial at \(x\) is returned.If 2: For even \(m\) when \(x\) is
None
, a univariate polynomial over the base ring of the curve is returned, which includes a factor of degree 3 whose roots are the \(x\)-coordinates of the \(2\)-torsion points. Similarly, when \(x\) is notNone
, the evaluation of such a polynomial at \(x\) is returned.If 1: For even \(m\) when \(x\) is
None
, a bivariate polynomial over the base ring of the curve is returned, which includes a factor \(2y+a_1x+a_3\) having simple zeros at the \(2\)-torsion points. When \(x\) is notNone
, it should be a tuple of length 2, and the evaluation of such a polynomial at \(x\) is returned.force_evaluate
(optional) – 0, 1, or 2By default, this method makes use of previously cached generic division polynomials to compute the value of the polynomial at a given element \(x\) whenever it appears beneficial to do so. Explicitly setting this flag overrides the default behavior.
Note that the complexity of evaluating a generic division polynomial scales much worse than that of computing the value at a point directly (using the recursive formulas), hence setting this flag can be detrimental to performance.
If 0: Do not use cached generic division polynomials.
If 1: If the generic division polynomial for this \(m\) has been cached before, evaluate it at \(x\) to compute the result.
If 2: Compute the value at \(x\) by evaluating the generic division polynomial. If the generic \(m\)-division polynomial has not yet been cached, compute and cache it first.
EXAMPLES:
sage: E = EllipticCurve([0,0,1,-1,0]) sage: E.division_polynomial(1) 1 sage: E.division_polynomial(2, two_torsion_multiplicity=0) 1 sage: E.division_polynomial(2, two_torsion_multiplicity=1) 2*y + 1 sage: E.division_polynomial(2, two_torsion_multiplicity=2) 4*x^3 - 4*x + 1 sage: E.division_polynomial(2) 4*x^3 - 4*x + 1 sage: [E.division_polynomial(3, two_torsion_multiplicity=i) for i in range(3)] [3*x^4 - 6*x^2 + 3*x - 1, 3*x^4 - 6*x^2 + 3*x - 1, 3*x^4 - 6*x^2 + 3*x - 1] sage: [type(E.division_polynomial(3, two_torsion_multiplicity=i)) for i in range(3)] [<... 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint'>, <... 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>, <... 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint'>]
sage: E = EllipticCurve([0, -1, 1, -10, -20]) sage: R.<z>=PolynomialRing(QQ) sage: E.division_polynomial(4,z,0) 2*z^6 - 4*z^5 - 100*z^4 - 790*z^3 - 210*z^2 - 1496*z - 5821 sage: E.division_polynomial(4,z) 8*z^9 - 24*z^8 - 464*z^7 - 2758*z^6 + 6636*z^5 + 34356*z^4 + 53510*z^3 + 99714*z^2 + 351024*z + 459859
This does not work, since when two_torsion_multiplicity is 1, we compute a bivariate polynomial, and must evaluate at a tuple of length 2:
sage: E.division_polynomial(4,z,1) Traceback (most recent call last): ... ValueError: x should be a tuple of length 2 (or None) when two_torsion_multiplicity is 1 sage: R.<z,w>=PolynomialRing(QQ,2) sage: E.division_polynomial(4,(z,w),1).factor() (2*w + 1) * (2*z^6 - 4*z^5 - 100*z^4 - 790*z^3 - 210*z^2 - 1496*z - 5821)
We can also evaluate this bivariate polynomial at a point:
sage: P = E(5,5) sage: E.division_polynomial(4,P,two_torsion_multiplicity=1) -1771561
- division_polynomial_0(n, x=None)#
Return the \(n^{th}\) torsion (division) polynomial, without the 2-torsion factor if \(n\) is even, as a polynomial in \(x\).
These are the polynomials \(g_n\) defined in [MT1991], but with the sign flipped for even \(n\), so that the leading coefficient is always positive.
Note
This function is intended for internal use; users should use
division_polynomial()
.See also
_multiple_x_numerator()
_multiple_x_denominator()
INPUT:
n
– positive integer, or the special values-1
and-2
which mean \(B_6 = (2y + a_1 x + a_3)^2\) and \(B_6^2\) respectively (in the notation of [MT1991]); or a list of integers.x
– a ring element to use as the “x” variable orNone
(default:None
). IfNone
, then a new polynomial ring will be constructed over the base ring of the elliptic curve, and its generator will be used asx
. Note thatx
does not need to be a generator of a polynomial ring; any ring element is ok. This permits fast calculation of the torsion polynomial evaluated on any element of a ring.
ALGORITHM:
Recursion described in [MT1991]. The recursive formulae are evaluated \(O(\log^2 n)\) times.
AUTHORS:
David Harvey (2006-09-24): initial version
John Cremona (2008-08-26): unified division polynomial code
EXAMPLES:
sage: E = EllipticCurve("37a") sage: E.division_polynomial_0(1) 1 sage: E.division_polynomial_0(2) 1 sage: E.division_polynomial_0(3) 3*x^4 - 6*x^2 + 3*x - 1 sage: E.division_polynomial_0(4) 2*x^6 - 10*x^4 + 10*x^3 - 10*x^2 + 2*x + 1 sage: E.division_polynomial_0(5) 5*x^12 - 62*x^10 + 95*x^9 - 105*x^8 - 60*x^7 + 285*x^6 - 174*x^5 - 5*x^4 - 5*x^3 + 35*x^2 - 15*x + 2 sage: E.division_polynomial_0(6) 3*x^16 - 72*x^14 + 168*x^13 - 364*x^12 + 1120*x^10 - 1144*x^9 + 300*x^8 - 540*x^7 + 1120*x^6 - 588*x^5 - 133*x^4 + 252*x^3 - 114*x^2 + 22*x - 1 sage: E.division_polynomial_0(7) 7*x^24 - 308*x^22 + 986*x^21 - 2954*x^20 + 28*x^19 + 17171*x^18 - 23142*x^17 + 511*x^16 - 5012*x^15 + 43804*x^14 - 7140*x^13 - 96950*x^12 + 111356*x^11 - 19516*x^10 - 49707*x^9 + 40054*x^8 - 124*x^7 - 18382*x^6 + 13342*x^5 - 4816*x^4 + 1099*x^3 - 210*x^2 + 35*x - 3 sage: E.division_polynomial_0(8) 4*x^30 - 292*x^28 + 1252*x^27 - 5436*x^26 + 2340*x^25 + 39834*x^24 - 79560*x^23 + 51432*x^22 - 142896*x^21 + 451596*x^20 - 212040*x^19 - 1005316*x^18 + 1726416*x^17 - 671160*x^16 - 954924*x^15 + 1119552*x^14 + 313308*x^13 - 1502818*x^12 + 1189908*x^11 - 160152*x^10 - 399176*x^9 + 386142*x^8 - 220128*x^7 + 99558*x^6 - 33528*x^5 + 6042*x^4 + 310*x^3 - 406*x^2 + 78*x - 5
sage: E.division_polynomial_0(18) % E.division_polynomial_0(6) == 0 True
An example to illustrate the relationship with torsion points:
sage: F = GF(11) sage: E = EllipticCurve(F, [0, 2]); E Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field of size 11 sage: f = E.division_polynomial_0(5); f 5*x^12 + x^9 + 8*x^6 + 4*x^3 + 7 sage: f.factor() (5) * (x^2 + 5) * (x^2 + 2*x + 5) * (x^2 + 5*x + 7) * (x^2 + 7*x + 7) * (x^2 + 9*x + 5) * (x^2 + 10*x + 7)
This indicates that the \(x\)-coordinates of all the 5-torsion points of \(E\) are in \(\GF{11^2}\), and therefore the \(y\)-coordinates are in \(\GF{11^4}\):
sage: K = GF(11^4, 'a') sage: X = E.change_ring(K) sage: f = X.division_polynomial_0(5) sage: x_coords = f.roots(multiplicities=False); x_coords [10*a^3 + 4*a^2 + 5*a + 6, 9*a^3 + 8*a^2 + 10*a + 8, 8*a^3 + a^2 + 4*a + 10, 8*a^3 + a^2 + 4*a + 8, 8*a^3 + a^2 + 4*a + 4, 6*a^3 + 9*a^2 + 3*a + 4, 5*a^3 + 2*a^2 + 8*a + 7, 3*a^3 + 10*a^2 + 7*a + 8, 3*a^3 + 10*a^2 + 7*a + 3, 3*a^3 + 10*a^2 + 7*a + 1, 2*a^3 + 3*a^2 + a + 7, a^3 + 7*a^2 + 6*a]
Now we check that these are exactly the \(x\)-coordinates of the 5-torsion points of \(E\):
sage: for x in x_coords: ....: assert X.lift_x(x).order() == 5
The roots of the polynomial are the \(x\)-coordinates of the points \(P\) such that \(mP=0\) but \(2P\not=0\):
sage: E = EllipticCurve('14a1') sage: T = E.torsion_subgroup() sage: [n*T.0 for n in range(6)] [(0 : 1 : 0), (9 : 23 : 1), (2 : 2 : 1), (1 : -1 : 1), (2 : -5 : 1), (9 : -33 : 1)] sage: pol = E.division_polynomial_0(6) sage: xlist = pol.roots(multiplicities=False); xlist [9, 2, -1/3, -5] sage: [E.lift_x(x, all=True) for x in xlist] [[(9 : 23 : 1), (9 : -33 : 1)], [(2 : 2 : 1), (2 : -5 : 1)], [], []]
Note
The point of order 2 and the identity do not appear. The points with \(x=-1/3\) and \(x=-5\) are not rational.
- formal()#
Return the formal group associated to this elliptic curve.
This method is cached.
EXAMPLES:
sage: E = EllipticCurve("37a") sage: E.formal_group() Formal Group associated to the Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
- formal_group()#
Return the formal group associated to this elliptic curve.
This method is cached.
EXAMPLES:
sage: E = EllipticCurve("37a") sage: E.formal_group() Formal Group associated to the Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
- gen(i)#
Function returning the i’th generator of this elliptic curve.
Note
Relies on gens() being implemented.
EXAMPLES:
sage: R.<a1,a2,a3,a4,a6>=QQ[] sage: E = EllipticCurve([a1,a2,a3,a4,a6]) sage: E.gen(0) Traceback (most recent call last): ... NotImplementedError: not implemented.
- gens()#
Placeholder function to return generators of an elliptic curve.
Note
This functionality is implemented in certain derived classes, such as EllipticCurve_rational_field.
EXAMPLES:
sage: R.<a1,a2,a3,a4,a6>=QQ[] sage: E = EllipticCurve([a1,a2,a3,a4,a6]) sage: E.gens() Traceback (most recent call last): ... NotImplementedError: not implemented. sage: E = EllipticCurve(QQ,[1,1]) sage: E.gens() [(0 : 1 : 1)]
- hyperelliptic_polynomials()#
Return a pair of polynomials \(g(x)\), \(h(x)\) such that this elliptic curve can be defined by the standard hyperelliptic equation
\[y^2 + h(x)y = g(x).\]EXAMPLES:
sage: R.<a1,a2,a3,a4,a6>=QQ[] sage: E = EllipticCurve([a1,a2,a3,a4,a6]) sage: E.hyperelliptic_polynomials() (x^3 + a2*x^2 + a4*x + a6, a1*x + a3)
- is_isomorphic(other, field=None)#
Return whether or not self is isomorphic to other.
INPUT:
other
– another elliptic curve.field
(default None) – a field into which the coefficients of the curves may be coerced (by default, uses the base field of the curves).
OUTPUT:
(bool) True if there is an isomorphism from curve
self
to curveother
defined overfield
.EXAMPLES:
sage: E = EllipticCurve('389a') sage: F = E.change_weierstrass_model([2,3,4,5]); F Elliptic Curve defined by y^2 + 4*x*y + 11/8*y = x^3 - 3/2*x^2 - 13/16*x over Rational Field sage: E.is_isomorphic(F) True sage: E.is_isomorphic(F.change_ring(CC)) False
- is_on_curve(x, y)#
Return True if \((x,y)\) is an affine point on this curve.
INPUT:
x
,y
– elements of the base ring of the curve.
EXAMPLES:
sage: E = EllipticCurve(QQ,[1,1]) sage: E.is_on_curve(0,1) True sage: E.is_on_curve(1,1) False
- is_x_coord(x)#
Return True if
x
is the \(x\)-coordinate of a point on this curve.Note
See also
lift_x()
to find the point(s) with a given \(x\)-coordinate. This function may be useful in cases where testing an element of the base field for being a square is faster than finding its square root.EXAMPLES:
sage: E = EllipticCurve('37a'); E Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field sage: E.is_x_coord(1) True sage: E.is_x_coord(2) True
There are no rational points with x-coordinate 3:
sage: E.is_x_coord(3) False
However, there are such points in \(E(\RR)\):
sage: E.change_ring(RR).is_x_coord(3) True
And of course it always works in \(E(\CC)\):
sage: E.change_ring(RR).is_x_coord(-3) False sage: E.change_ring(CC).is_x_coord(-3) True
AUTHORS:
John Cremona (2008-08-07): adapted from
lift_x()
- isomorphism_to(other)#
Given another weierstrass model
other
of self, return an isomorphism from self toother
.INPUT:
other
– an elliptic curve isomorphic toself
.
OUTPUT:
(Weierstrassmorphism) An isomorphism from self to other.
Note
If the curves in question are not isomorphic, a
ValueError
is raised.EXAMPLES:
sage: E = EllipticCurve('37a') sage: F = E.short_weierstrass_model() sage: w = E.isomorphism_to(F); w Elliptic-curve morphism: From: Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field To: Elliptic Curve defined by y^2 = x^3 - 16*x + 16 over Rational Field Via: (u,r,s,t) = (1/2, 0, 0, -1/2) sage: P = E(0,-1,1) sage: w(P) (0 : -4 : 1) sage: w(5*P) (1 : 1 : 1) sage: 5*w(P) (1 : 1 : 1) sage: 120*w(P) == w(120*P) True
We can also handle injections to different base rings:
sage: K.<a> = NumberField(x^3-7) sage: E.isomorphism_to(E.change_ring(K)) Elliptic-curve morphism: From: Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field To: Elliptic Curve defined by y^2 + y = x^3 + (-1)*x over Number Field in a with defining polynomial x^3 - 7 Via: (u,r,s,t) = (1, 0, 0, 0)
- isomorphisms(other, field=None)#
Return the set of isomorphisms from self to other (as a list).
INPUT:
other
– another elliptic curve.field
(defaultNone
) – a field into which the coefficients of the curves may be coerced (by default, uses the base field of the curves).
OUTPUT:
(list) A list of
WeierstrassIsomorphism
objects consisting of all the isomorphisms from the curveself
to the curveother
defined overfield
.EXAMPLES:
sage: E = EllipticCurve_from_j(QQ(0)) # a curve with j=0 over QQ sage: F = EllipticCurve('27a3') # should be the same one sage: E.isomorphisms(F); [Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Rational Field Via: (u,r,s,t) = (-1, 0, 0, -1), Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Rational Field Via: (u,r,s,t) = (1, 0, 0, 0)]
We can also find isomorphisms defined over extension fields:
sage: E = EllipticCurve(GF(7),[0,0,0,1,1]) sage: F = EllipticCurve(GF(7),[0,0,0,1,-1]) sage: E.isomorphisms(F) [] sage: E.isomorphisms(F,GF(49,'a')) [Elliptic-curve morphism: From: Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field in a of size 7^2 To: Elliptic Curve defined by y^2 = x^3 + x + 6 over Finite Field in a of size 7^2 Via: (u,r,s,t) = (a + 3, 0, 0, 0), Elliptic-curve morphism: From: Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field in a of size 7^2 To: Elliptic Curve defined by y^2 = x^3 + x + 6 over Finite Field in a of size 7^2 Via: (u,r,s,t) = (6*a + 4, 0, 0, 0)]
- j_invariant()#
Return the \(j\)-invariant of this elliptic curve.
This method is cached.
EXAMPLES:
sage: E = EllipticCurve([0,0,1,-1,0]) sage: E.j_invariant() 110592/37 sage: E = EllipticCurve([0, -1, 1, -10, -20]) sage: E.j_invariant() -122023936/161051 sage: E = EllipticCurve([-4,0]) sage: E.j_invariant() 1728 sage: E = EllipticCurve([GF(7)(2),1]) sage: E.j_invariant() 1
- lift_x(x, all=False, extend=False)#
Return one or all points with given \(x\)-coordinate.
INPUT:
x
– an element of the base ring of the curve, or of an extension.all
(bool, default False) – if True, return a (possibly empty) list of all points; if False, return just one point, or raise a ValueError if there are none.extend
(bool, default False) –if
False
, extend the base if necessary and possible to include \(x\), and only return point(s) defined over this ring, or raise an error when there are none with this \(x\)-coordinate;If
True
, the base ring will be extended if necessary to contain the \(y\)-coordinates of the point(s) with this \(x\)-coordinate, in addition to a possible base change to include \(x\).
OUTPUT:
A point or list of up to 2 points on this curve, or a base-change of this curve to a larger ring.
See also
EXAMPLES:
sage: E = EllipticCurve('37a'); E Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field sage: E.lift_x(1) (1 : 0 : 1) sage: E.lift_x(2) (2 : 2 : 1) sage: E.lift_x(1/4, all=True) [(1/4 : -3/8 : 1), (1/4 : -5/8 : 1)]
There are no rational points with \(x\)-coordinate 3:
sage: E.lift_x(3) Traceback (most recent call last): ... ValueError: No point with x-coordinate 3 on Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
We can use the
extend
parameter to make the necessary quadratic extension. Note that in such cases the returned point is a point on a new curve object, the result of changing the base ring to the parent of \(x\):sage: P = E.lift_x(3, extend=True); P (3 : y : 1) sage: P.curve() Elliptic Curve defined by y^2 + y = x^3 + (-1)*x over Number Field in y with defining polynomial y^2 + y - 24
Or we can extend scalars. There are two such points in \(E(\RR)\):
sage: E.change_ring(RR).lift_x(3, all=True) [(3.00000000000000 : 4.42442890089805 : 1.00000000000000), (3.00000000000000 : -5.42442890089805 : 1.00000000000000)]
And of course it always works in \(E(\CC)\):
sage: E.change_ring(RR).lift_x(.5, all=True) [] sage: E.change_ring(CC).lift_x(.5) (0.500000000000000 : -0.500000000000000 + 0.353553390593274*I : 1.00000000000000)
In this example we start with a curve defined over \(\QQ\) which has no rational points with \(x=0\), but using
extend = True
we can construct such a point over a quadratic field:sage: E = EllipticCurve([0,0,0,0,2]); E Elliptic Curve defined by y^2 = x^3 + 2 over Rational Field sage: P = E.lift_x(0, extend=True); P (0 : y : 1) sage: P.curve() Elliptic Curve defined by y^2 = x^3 + 2 over Number Field in y with defining polynomial y^2 - 2
We can perform these operations over finite fields too:
sage: E = EllipticCurve('37a').change_ring(GF(17)); E Elliptic Curve defined by y^2 + y = x^3 + 16*x over Finite Field of size 17 sage: E.lift_x(7) (7 : 11 : 1) sage: E.lift_x(3) Traceback (most recent call last): ... ValueError: No point with x-coordinate 3 on Elliptic Curve defined by y^2 + y = x^3 + 16*x over Finite Field of size 17
Note that there is only one lift with \(x\)-coordinate 10 in \(E(\GF{17})\):
sage: E.lift_x(10, all=True) [(10 : 8 : 1)]
We can lift over more exotic rings too. If the supplied x value is in an extension of the base, note that the point returned is on the base-extended curve:
sage: E = EllipticCurve('37a') sage: P = E.lift_x(pAdicField(17, 5)(6)); P (6 + O(17^5) : 2 + 16*17 + 16*17^2 + 16*17^3 + 16*17^4 + O(17^5) : 1 + O(17^5)) sage: P.curve() Elliptic Curve defined by y^2 + (1+O(17^5))*y = x^3 + (16+16*17+16*17^2+16*17^3+16*17^4+O(17^5))*x over 17-adic Field with capped relative precision 5 sage: K.<t> = PowerSeriesRing(QQ, 't', 5) sage: P = E.lift_x(1+t); P (1 + t : 2*t - t^2 + 5*t^3 - 21*t^4 + O(t^5) : 1) sage: K.<a> = GF(16) sage: P = E.change_ring(K).lift_x(a^3); P (a^3 : a^3 + a : 1) sage: P.curve() Elliptic Curve defined by y^2 + y = x^3 + x over Finite Field in a of size 2^4
We can extend the base field to include the associated \(y\) value(s):
sage: E = EllipticCurve([0,0,0,0,2]); E Elliptic Curve defined by y^2 = x^3 + 2 over Rational Field sage: x = polygen(QQ) sage: P = E.lift_x(x, extend=True); P (x : y : 1)
This point is a generic point on E:
sage: P.curve() Elliptic Curve defined by y^2 = x^3 + 2 over Univariate Quotient Polynomial Ring in y over Fraction Field of Univariate Polynomial Ring in x over Rational Field with modulus y^2 - x^3 - 2 sage: -P (x : -y : 1) sage: 2*P ((1/4*x^4 - 4*x)/(x^3 + 2) : ((1/8*x^6 + 5*x^3 - 4)/(x^6 + 4*x^3 + 4))*y : 1)
Check that trac ticket #30297 is fixed:
sage: K = Qp(5) sage: E = EllipticCurve([K(0), K(1)]) sage: E.lift_x(1, extend=True) (1 + O(5^20) : y + O(5^20) : 1 + O(5^20))
AUTHORS:
Robert Bradshaw (2007-04-24)
John Cremona (2017-11-10)
- montgomery_model(twisted=False, morphism=False)#
Return a (twisted or untwisted) Montgomery model for this elliptic curve, if possible.
A Montgomery curve is a smooth projective curve of the form
\[BY^2 = X^3 + AX^2 + X.\]The Montgomery curve is called untwisted if \(B=1\).
INPUT:
twisted
– boolean (default:False
); allow \(B \neq 1\)morphism
– boolean (default:False
); also return an isomorphism from this curve to the computed Montgomery model
OUTPUT:
If
twisted
isFalse
(the default), anEllipticCurve_generic
object encapsulating an untwisted Montgomery curve. Otherwise, aProjectivePlaneCurve
object encapsulating a (potentially twisted) Montgomery curve.If
morphism
isTrue
, this method returns a tuple consisting of such a curve together with an isomorphism of suitable type (eitherWeierstrassIsomorphism
orWeierstrassTransformationWithInverse
) from this curve to the Montgomery model.EXAMPLES:
sage: E = EllipticCurve(QQbar, '11a1') sage: E.montgomery_model() Elliptic Curve defined by y^2 = x^3 + (-1.953522420987248?)*x^2 + x over Algebraic Field
sage: E = EllipticCurve(GF(431^2), [7,7]) sage: E.montgomery_model() Elliptic Curve defined by y^2 = x^3 + (51*z2+190)*x^2 + x over Finite Field in z2 of size 431^2
An isomorphism between the Montgomery and Weierstrass form can be obtained using the
morphism
parameter:sage: E.montgomery_model(morphism=True) (Elliptic Curve defined by y^2 = x^3 + (51*z2+190)*x^2 + x over Finite Field in z2 of size 431^2, Elliptic-curve morphism: From: Elliptic Curve defined by y^2 = x^3 + 7*x + 7 over Finite Field in z2 of size 431^2 To: Elliptic Curve defined by y^2 = x^3 + (51*z2+190)*x^2 + x over Finite Field in z2 of size 431^2 Via: (u,r,s,t) = (64*z2 + 407, 159, 0, 0))
Not all elliptic curves have a Montgomery model over their field of definition:
sage: E = EllipticCurve(GF(257), [1,1]) sage: E.montgomery_model() Traceback (most recent call last): ... ValueError: Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 257 has no Montgomery model
sage: E = EllipticCurve(GF(257), [10,10]) sage: E.montgomery_model() Traceback (most recent call last): ... ValueError: Elliptic Curve defined by y^2 = x^3 + 10*x + 10 over Finite Field of size 257 has no untwisted Montgomery model
However, as hinted by the error message, the latter curve does admit a twisted Montgomery model, which can be computed by passing
twisted=True
:sage: E.montgomery_model(twisted=True) Projective Plane Curve over Finite Field of size 257 defined by -x^3 + 8*x^2*z - 127*y^2*z - x*z^2
Since Sage internally represents elliptic curves as (long) Weierstrass curves, which do not feature the Montgomery \(B\) coefficient, the returned curve in this case is merely a
ProjectivePlaneCurve
rather than the usualEllipticCurve_generic
.Arithmetic on curves of this type is not implemented natively, but can easily be emulated by mapping back and forth to the corresponding Weierstrass curve:
sage: C, f = E.montgomery_model(twisted=True, morphism=True) sage: f Scheme morphism: From: Elliptic Curve defined by y^2 = x^3 + 10*x + 10 over Finite Field of size 257 To: Projective Plane Curve over Finite Field of size 257 defined by -x^3 + 8*x^2*z - 127*y^2*z - x*z^2 Defn: Defined on coordinates by sending (x : y : z) to (x + 116*z : -y : -85*z) sage: g = f.inverse(); g Scheme morphism: From: Projective Plane Curve over Finite Field of size 257 defined by -x^3 + 8*x^2*z - 127*y^2*z - x*z^2 To: Elliptic Curve defined by y^2 = x^3 + 10*x + 10 over Finite Field of size 257 Defn: Defined on coordinates by sending (x : y : z) to (-85*x - 116*z : 85*y : z) sage: P = C(70, 8) sage: Q = C(17, 17) sage: P + Q # this doesn't work... Traceback (most recent call last): ... TypeError: unsupported operand parent(s) for +: ... sage: f(g(P) + g(Q)) # ...but this does (107 : 168 : 1)
Using the fact that the Weil pairing satisfies \(e(\psi(P),\psi(Q)) = e(P,Q)^{\deg\psi}\), even pairings can be emulated in this way (note that isomorphisms have degree \(1\)):
sage: F.<z2> = GF(257^2) sage: C_ = C.change_ring(F) sage: g_ = g.change_ring(F) sage: g_(P).order() 12 sage: T = C_(-7 * z2 - 57, 31 * z2 - 52, 1) sage: g_(T).order() 12 sage: g_(P).weil_pairing(g_(T), 12) 15*z2 + 204
Another alternative is to simply extend the base field enough for the curve to have an untwisted Montgomery model:
sage: C_ = E.change_ring(F).montgomery_model(); C_ Elliptic Curve defined by y^2 = x^3 + 249*x^2 + x over Finite Field in z2 of size 257^2 sage: h = C.defining_polynomial().change_ring(F); h -x^3 + 8*x^2*z - 127*y^2*z - x*z^2 sage: C_.is_isomorphic(EllipticCurve_from_cubic(h).codomain()) True
See also
The inverse conversion — computing a Weierstrass model for a given Montgomery curve — can be performed using
EllipticCurve_from_cubic()
.ALGORITHM: [CS2018], §2.4
REFERENCES:
Original publication: [Mont1987], §10.3.1
More recent survey article: [CS2018]
- multiplication_by_m(m, x_only=False)#
Return the multiplication-by-\(m\) map from
self
toself
The result is a pair of rational functions in two variables \(x\), \(y\) (or a rational function in one variable \(x\) if
x_only
isTrue
).INPUT:
m
– a nonzero integerx_only
– boolean (default:False
) ifTrue
, return only the \(x\)-coordinate of the map (as a rational function in one variable).
OUTPUT:
a pair \((f(x), g(x,y))\), where \(f\) and \(g\) are rational functions with the degree of \(y\) in \(g(x,y)\) exactly 1,
or just \(f(x)\) if
x_only
isTrue
Note
The result is not cached.
m
is allowed to be negative (but not 0).
EXAMPLES:
sage: E = EllipticCurve([-1,3])
We verify that multiplication by 1 is just the identity:
sage: E.multiplication_by_m(1) (x, y)
Multiplication by 2 is more complicated:
sage: f = E.multiplication_by_m(2) sage: f ((x^4 + 2*x^2 - 24*x + 1)/(4*x^3 - 4*x + 12), (8*x^6*y - 40*x^4*y + 480*x^3*y - 40*x^2*y + 96*x*y - 568*y)/(64*x^6 - 128*x^4 + 384*x^3 + 64*x^2 - 384*x + 576))
Grab only the x-coordinate (less work):
sage: mx = E.multiplication_by_m(2, x_only=True); mx (1/4*x^4 + 1/2*x^2 - 6*x + 1/4)/(x^3 - x + 3) sage: mx.parent() Fraction Field of Univariate Polynomial Ring in x over Rational Field
We check that it works on a point:
sage: P = E([2,3]) sage: eval = lambda f,P: [fi(P[0],P[1]) for fi in f] sage: assert E(eval(f,P)) == 2*P
We do the same but with multiplication by 3:
sage: f = E.multiplication_by_m(3) sage: assert E(eval(f,P)) == 3*P
And the same with multiplication by 4:
sage: f = E.multiplication_by_m(4) sage: assert E(eval(f,P)) == 4*P
And the same with multiplication by -1,-2,-3,-4:
sage: for m in [-1,-2,-3,-4]: ....: f = E.multiplication_by_m(m) ....: assert E(eval(f,P)) == m*P
- multiplication_by_m_isogeny(m)#
Return the
EllipticCurveIsogeny
object associated to the multiplication-by-\(m\) map on this elliptic curve.The resulting isogeny will have the associated rational maps (i.e. those returned by \(self.multiplication_by_m()\)) already computed.
NOTE: This function is currently much slower than the result of
self.multiplication_by_m()
, because constructing an isogeny precomputes a significant amount of information. See trac ticket #7368 and trac ticket #8014 for the status of improving this situation.INPUT:
m
– a nonzero integer
OUTPUT:
An
EllipticCurveIsogeny
object associated to the multiplication-by-\(m\) map on this elliptic curve.
EXAMPLES:
sage: E = EllipticCurve('11a1') sage: E.multiplication_by_m_isogeny(7) Isogeny of degree 49 from Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
- pari_curve()#
Return the PARI curve corresponding to this elliptic curve.
The result is cached.
EXAMPLES:
sage: E = EllipticCurve([RR(0), RR(0), RR(1), RR(-1), RR(0)]) sage: e = E.pari_curve() sage: type(e) <... 'cypari2.gen.Gen'> sage: e.type() 't_VEC' sage: e.disc() 37.0000000000000
Over a finite field:
sage: EllipticCurve(GF(41),[2,5]).pari_curve() [Mod(0, 41), Mod(0, 41), Mod(0, 41), Mod(2, 41), Mod(5, 41), Mod(0, 41), Mod(4, 41), Mod(20, 41), Mod(37, 41), Mod(27, 41), Mod(26, 41), Mod(4, 41), Mod(11, 41), Vecsmall([3]), [41, [9, 31, [6, 0, 0, 0]]], [0, 0, 0, 0]]
Over a \(p\)-adic field:
sage: Qp = pAdicField(5, prec=3) sage: E = EllipticCurve(Qp,[3, 4]) sage: E.pari_curve() [0, 0, 0, 3, 4, 0, 6, 16, -9, -144, -3456, -8640, 1728/5, Vecsmall([2]), [O(5^3)], [0, 0]] sage: E.j_invariant() 3*5^-1 + O(5)
Over a number field:
sage: K.<a> = QuadraticField(2) sage: E = EllipticCurve([1,a]) sage: E.pari_curve() [Mod(0, y^2 - 2), Mod(0, y^2 - 2), Mod(0, y^2 - 2), Mod(1, y^2 - 2), Mod(y, y^2 - 2), Mod(0, y^2 - 2), Mod(2, y^2 - 2), Mod(4*y, y^2 - 2), Mod(-1, y^2 - 2), Mod(-48, y^2 - 2), Mod(-864*y, y^2 - 2), Mod(-928, y^2 - 2), Mod(3456/29, y^2 - 2), Vecsmall([5]), [[y^2 - 2, [2, 0], 8, 1, [[1, -1.41421356237310; 1, 1.41421356237310], [1, -1.41421356237310; 1, 1.41421356237310], [16, -23; 16, 23], [2, 0; 0, 4], [4, 0; 0, 2], [2, 0; 0, 1], [2, [0, 2; 1, 0]], [2]], [-1.41421356237310, 1.41421356237310], [1, y], [1, 0; 0, 1], [1, 0, 0, 2; 0, 1, 1, 0]]], [0, 0, 0, 0, 0]]
PARI no longer requires that the \(j\)-invariant has negative \(p\)-adic valuation:
sage: E = EllipticCurve(Qp,[1, 1]) sage: E.j_invariant() # the j-invariant is a p-adic integer 2 + 4*5^2 + O(5^3) sage: E.pari_curve() [0, 0, 0, 1, 1, 0, 2, 4, -1, -48, -864, -496, 6912/31, Vecsmall([2]), [O(5^3)], [0, 0]]
- plot(xmin=None, xmax=None, components='both', **args)#
Draw a graph of this elliptic curve.
The plot method is only implemented when there is a natural coercion from the base ring of
self
toRR
. In this case,self
is plotted as if it was defined overRR
.INPUT:
xmin, xmax
– (optional) points will be computed at least within this range, but possibly farther.components
– a string, one of the following:both
– (default), scale so that both bounded and unbounded components appearbounded
– scale the plot to show the bounded component. Raises an error if there is only one real component.unbounded
– scale the plot to show the unbounded component, including the two flex points.
plot_points
– passed tosage.plot.generate_plot_points()
adaptive_tolerance
– passed tosage.plot.generate_plot_points()
adaptive_recursion
– passed tosage.plot.generate_plot_points()
randomize
– passed tosage.plot.generate_plot_points()
**args
– all other options are passed tosage.plot.line.Line
EXAMPLES:
sage: E = EllipticCurve([0,-1]) sage: plot(E, rgbcolor=hue(0.7)) Graphics object consisting of 1 graphics primitive sage: E = EllipticCurve('37a') sage: plot(E) Graphics object consisting of 2 graphics primitives sage: plot(E, xmin=25,xmax=26) Graphics object consisting of 2 graphics primitives
With trac ticket #12766 we added the components keyword:
sage: E.real_components() 2 sage: E.plot(components='bounded') Graphics object consisting of 1 graphics primitive sage: E.plot(components='unbounded') Graphics object consisting of 1 graphics primitive
If there is only one component then specifying components=’bounded’ raises a ValueError:
sage: E = EllipticCurve('9990be2') sage: E.plot(components='bounded') Traceback (most recent call last): ... ValueError: no bounded component for this curve
An elliptic curve defined over the Complex Field can not be plotted:
sage: E = EllipticCurve(CC, [0,0,1,-1,0]) sage: E.plot() Traceback (most recent call last): ... NotImplementedError: plotting of curves over Complex Field with 53 bits of precision is not implemented yet
- rst_transform(r, s, t)#
Return the transform of the curve by \((r,s,t)\) (with \(u=1\)).
INPUT:
r
,s
,t
– three elements of the base ring.
OUTPUT:
The elliptic curve obtained from self by the standard Weierstrass transformation \((u,r,s,t)\) with \(u=1\).
Note
This is just a special case of
change_weierstrass_model()
, with \(u=1\).EXAMPLES:
sage: R.<r,s,t>=QQ[] sage: E = EllipticCurve([1,2,3,4,5]) sage: E.rst_transform(r,s,t) Elliptic Curve defined by y^2 + (2*s+1)*x*y + (r+2*t+3)*y = x^3 + (-s^2+3*r-s+2)*x^2 + (3*r^2-r*s-2*s*t+4*r-3*s-t+4)*x + (r^3+2*r^2-r*t-t^2+4*r-3*t+5) over Multivariate Polynomial Ring in r, s, t over Rational Field
- scale_curve(u)#
Return the transform of the curve by scale factor \(u\).
INPUT:
u
– an invertible element of the base ring.
OUTPUT:
The elliptic curve obtained from self by the standard Weierstrass transformation \((u,r,s,t)\) with \(r=s=t=0\).
Note
This is just a special case of
change_weierstrass_model()
, with \(r=s=t=0\).EXAMPLES:
sage: K = Frac(PolynomialRing(QQ,'u')) sage: u = K.gen() sage: E = EllipticCurve([1,2,3,4,5]) sage: E.scale_curve(u) Elliptic Curve defined by y^2 + u*x*y + 3*u^3*y = x^3 + 2*u^2*x^2 + 4*u^4*x + 5*u^6 over Fraction Field of Univariate Polynomial Ring in u over Rational Field
- short_weierstrass_model(complete_cube=True)#
Return a short Weierstrass model for self.
INPUT:
complete_cube
– boolean (default: True); for meaning, see below.
OUTPUT:
An elliptic curve.
If
complete_cube=True
: Return a model of the form \(y^2 = x^3 + a*x + b\) for this curve. The characteristic must not be 2; in characteristic 3, it is only possible if \(b_2=0\).If
complete_cube=False
: Return a model of the form \(y^2 = x^3 + ax^2 + bx + c\) for this curve. The characteristic must not be 2.EXAMPLES:
sage: E = EllipticCurve([1,2,3,4,5]) sage: E Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field sage: F = E.short_weierstrass_model() sage: F Elliptic Curve defined by y^2 = x^3 + 4941*x + 185166 over Rational Field sage: E.is_isomorphic(F) True sage: F = E.short_weierstrass_model(complete_cube=False) sage: F Elliptic Curve defined by y^2 = x^3 + 9*x^2 + 88*x + 464 over Rational Field sage: E.is_isomorphic(F) True
sage: E = EllipticCurve(GF(3),[1,2,3,4,5]) sage: E.short_weierstrass_model(complete_cube=False) Elliptic Curve defined by y^2 = x^3 + x + 2 over Finite Field of size 3
This used to be different see trac ticket #3973:
sage: E.short_weierstrass_model() Elliptic Curve defined by y^2 = x^3 + x + 2 over Finite Field of size 3
More tests in characteristic 3:
sage: E = EllipticCurve(GF(3),[0,2,1,2,1]) sage: E.short_weierstrass_model() Traceback (most recent call last): ... ValueError: short_weierstrass_model(): no short model for Elliptic Curve defined by y^2 + y = x^3 + 2*x^2 + 2*x + 1 over Finite Field of size 3 (characteristic is 3) sage: E.short_weierstrass_model(complete_cube=False) Elliptic Curve defined by y^2 = x^3 + 2*x^2 + 2*x + 2 over Finite Field of size 3 sage: E.short_weierstrass_model(complete_cube=False).is_isomorphic(E) True
- torsion_polynomial(m, x=None, two_torsion_multiplicity=2, force_evaluate=None)#
Return the \(m^{th}\) division polynomial of this elliptic curve evaluated at \(x\).
The division polynomial is cached if \(x\) is
None
.INPUT:
m
– positive integer.x
– optional ring element to use as the \(x\) variable. If \(x\) isNone
(omitted), then a new polynomial ring will be constructed over the base ring of the elliptic curve, and its generator will be used as \(x\). Note that \(x\) does not need to be a generator of a polynomial ring; any ring element works. This permits fast calculation of the torsion polynomial evaluated on any element of a ring.two_torsion_multiplicity
– 0, 1, or 2If 0: For even \(m\) when \(x\) is
None
, a univariate polynomial over the base ring of the curve is returned, which omits factors whose roots are the \(x\)-coordinates of the \(2\)-torsion points. When \(x\) is notNone
, the evaluation of such a polynomial at \(x\) is returned.If 2: For even \(m\) when \(x\) is
None
, a univariate polynomial over the base ring of the curve is returned, which includes a factor of degree 3 whose roots are the \(x\)-coordinates of the \(2\)-torsion points. Similarly, when \(x\) is notNone
, the evaluation of such a polynomial at \(x\) is returned.If 1: For even \(m\) when \(x\) is
None
, a bivariate polynomial over the base ring of the curve is returned, which includes a factor \(2y+a_1x+a_3\) having simple zeros at the \(2\)-torsion points. When \(x\) is notNone
, it should be a tuple of length 2, and the evaluation of such a polynomial at \(x\) is returned.force_evaluate
(optional) – 0, 1, or 2By default, this method makes use of previously cached generic division polynomials to compute the value of the polynomial at a given element \(x\) whenever it appears beneficial to do so. Explicitly setting this flag overrides the default behavior.
Note that the complexity of evaluating a generic division polynomial scales much worse than that of computing the value at a point directly (using the recursive formulas), hence setting this flag can be detrimental to performance.
If 0: Do not use cached generic division polynomials.
If 1: If the generic division polynomial for this \(m\) has been cached before, evaluate it at \(x\) to compute the result.
If 2: Compute the value at \(x\) by evaluating the generic division polynomial. If the generic \(m\)-division polynomial has not yet been cached, compute and cache it first.
EXAMPLES:
sage: E = EllipticCurve([0,0,1,-1,0]) sage: E.division_polynomial(1) 1 sage: E.division_polynomial(2, two_torsion_multiplicity=0) 1 sage: E.division_polynomial(2, two_torsion_multiplicity=1) 2*y + 1 sage: E.division_polynomial(2, two_torsion_multiplicity=2) 4*x^3 - 4*x + 1 sage: E.division_polynomial(2) 4*x^3 - 4*x + 1 sage: [E.division_polynomial(3, two_torsion_multiplicity=i) for i in range(3)] [3*x^4 - 6*x^2 + 3*x - 1, 3*x^4 - 6*x^2 + 3*x - 1, 3*x^4 - 6*x^2 + 3*x - 1] sage: [type(E.division_polynomial(3, two_torsion_multiplicity=i)) for i in range(3)] [<... 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint'>, <... 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>, <... 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint'>]
sage: E = EllipticCurve([0, -1, 1, -10, -20]) sage: R.<z>=PolynomialRing(QQ) sage: E.division_polynomial(4,z,0) 2*z^6 - 4*z^5 - 100*z^4 - 790*z^3 - 210*z^2 - 1496*z - 5821 sage: E.division_polynomial(4,z) 8*z^9 - 24*z^8 - 464*z^7 - 2758*z^6 + 6636*z^5 + 34356*z^4 + 53510*z^3 + 99714*z^2 + 351024*z + 459859
This does not work, since when two_torsion_multiplicity is 1, we compute a bivariate polynomial, and must evaluate at a tuple of length 2:
sage: E.division_polynomial(4,z,1) Traceback (most recent call last): ... ValueError: x should be a tuple of length 2 (or None) when two_torsion_multiplicity is 1 sage: R.<z,w>=PolynomialRing(QQ,2) sage: E.division_polynomial(4,(z,w),1).factor() (2*w + 1) * (2*z^6 - 4*z^5 - 100*z^4 - 790*z^3 - 210*z^2 - 1496*z - 5821)
We can also evaluate this bivariate polynomial at a point:
sage: P = E(5,5) sage: E.division_polynomial(4,P,two_torsion_multiplicity=1) -1771561
- two_division_polynomial(x=None)#
Return the 2-division polynomial of this elliptic curve evaluated at
x
.INPUT:
x
– optional ring element to use as the \(x\) variable. Ifx
isNone
, then a new polynomial ring will be constructed over the base ring of the elliptic curve, and its generator will be used asx
. Note thatx
does not need to be a generator of a polynomial ring; any ring element is acceptable. This permits fast calculation of the torsion polynomial evaluated on any element of a ring.
EXAMPLES:
sage: E = EllipticCurve('5077a1') sage: E.two_division_polynomial() 4*x^3 - 28*x + 25 sage: E = EllipticCurve(GF(3^2,'a'),[1,1,1,1,1]) sage: E.two_division_polynomial() x^3 + 2*x^2 + 2 sage: E.two_division_polynomial().roots() [(2, 1), (2*a, 1), (a + 2, 1)]
- sage.schemes.elliptic_curves.ell_generic.is_EllipticCurve(x)#
Utility function to test if
x
is an instance of an Elliptic Curve class.EXAMPLES:
sage: from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve sage: E = EllipticCurve([1,2,3/4,7,19]) sage: is_EllipticCurve(E) True sage: is_EllipticCurve(0) False