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)[] # optional - sage.rings.finite_rings
sage: S.<v> = R[] # optional - sage.rings.finite_rings
sage: T = S.fraction_field() # optional - sage.rings.finite_rings
sage: E = EllipticCurve(T, [a, b]); E # optional - sage.rings.finite_rings
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) # optional - sage.rings.finite_rings
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:
WithEqualityById
,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]) # optional - sage.rings.finite_rings sage: E.a_invariants() # optional - sage.rings.finite_rings (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]) # optional - sage.rings.finite_rings sage: E.a_invariants() # optional - sage.rings.finite_rings (0, 0, 0, 3, 5)
- automorphisms(field=None)#
Return the set of isomorphisms from
self
to itself (as a list).The identity and negation morphisms are guaranteed to appear as the first and second entry of the returned 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, 0), 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)]
We can also find automorphisms defined over extension fields:
sage: x = polygen(ZZ, 'x') sage: K.<a> = NumberField(x^2 + 3) # adjoin roots of unity # optional - sage.rings.number_field sage: E.automorphisms(K) # optional - sage.rings.number_field [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), 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) = (-1/2*a - 1/2, 0, 0, 0), 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) = (1/2*a + 1/2, 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) = (1/2*a - 1/2, 0, 0, 0), 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) = (-1/2*a + 1/2, 0, 0, -1)]
sage: [len(EllipticCurve_from_j(GF(q,'a')(0)).automorphisms()) # optional - sage.rings.finite_rings ....: 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 # optional - sage.rings.finite_rings 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 # optional - sage.rings.finite_rings 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]) # optional - sage.rings.finite_rings sage: E.base_ring() # optional - sage.rings.finite_rings 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() # optional - sage.rings.finite_rings sage: F4 = GF(5^4,'b'); b = F4.gen() # optional - sage.rings.finite_rings sage: roots = a.charpoly().roots(ring=F4, multiplicities=False) # optional - sage.rings.finite_rings sage: h = F2.hom([roots[0]], F4) # optional - sage.rings.finite_rings sage: E = EllipticCurve(F2, [1,a]); E # optional - sage.rings.finite_rings Elliptic Curve defined by y^2 = x^3 + x + a over Finite Field in a of size 5^2 sage: E.change_ring(h) # optional - sage.rings.finite_rings 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]) # optional - sage.rings.finite_rings sage: E.discriminant() # optional - sage.rings.finite_rings 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) # optional - sage.rings.finite_rings sage: E = EllipticCurve(F, [0, 2]); E # optional - sage.rings.finite_rings Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field of size 11 sage: f = E.division_polynomial_0(5); f # optional - sage.rings.finite_rings 5*x^12 + x^9 + 8*x^6 + 4*x^3 + 7 sage: f.factor() # optional - sage.rings.finite_rings (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') # optional - sage.rings.finite_rings sage: X = E.change_ring(K) # optional - sage.rings.finite_rings sage: f = X.division_polynomial_0(5) # optional - sage.rings.finite_rings sage: x_coords = f.roots(multiplicities=False); x_coords # optional - sage.rings.finite_rings [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: # optional - sage.rings.finite_rings ....: 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 : -33 : 1), (9 : 23 : 1)], [(2 : -5 : 1), (2 : 2 : 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
- frobenius_isogeny(n=1)#
Return the \(n\)-power Frobenius isogeny from this curve to its Galois conjugate.
The Frobenius endomorphism is the special case where \(n\) is divisible by the degree of the base ring of the curve.
See also
EXAMPLES:
sage: z3, = GF(13^3).gens() # optional - sage.rings.finite_rings sage: E = EllipticCurve([z3, z3^2]) # optional - sage.rings.finite_rings sage: E.frobenius_isogeny() # optional - sage.rings.finite_rings Frobenius isogeny of degree 13: From: Elliptic Curve defined by y^2 = x^3 + z3*x + z3^2 over Finite Field in z3 of size 13^3 To: Elliptic Curve defined by y^2 = x^3 + (5*z3^2+7*z3+11)*x + (5*z3^2+12*z3+1) over Finite Field in z3 of size 13^3 sage: E.frobenius_isogeny(3) # optional - sage.rings.finite_rings Frobenius endomorphism of degree 2197 = 13^3: From: Elliptic Curve defined by y^2 = x^3 + z3*x + z3^2 over Finite Field in z3 of size 13^3 To: Elliptic Curve defined by y^2 = x^3 + z3*x + z3^2 over Finite Field in z3 of size 13^3
- 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 toother
.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
ofself
, return an isomorphism fromself
toother
.INPUT:
other
– an elliptic curve isomorphic toself
.
OUTPUT:
(Weierstrassmorphism) An isomorphism from
self
toother
.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: x = polygen(ZZ, 'x') sage: K.<a> = NumberField(x^3 - 7) # optional - sage.rings.number_field sage: E.isomorphism_to(E.change_ring(K)) # optional - sage.rings.number_field 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
toother
(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, 0), 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)]
We can also find isomorphisms defined over extension fields:
sage: E = EllipticCurve(GF(7), [0,0,0,1,1]) # optional - sage.rings.finite_rings sage: F = EllipticCurve(GF(7), [0,0,0,1,-1]) # optional - sage.rings.finite_rings sage: E.isomorphisms(F) # optional - sage.rings.finite_rings [] sage: E.isomorphisms(F, GF(49,'a')) # optional - sage.rings.finite_rings [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]) # optional - sage.rings.finite_rings sage: E.j_invariant() # optional - sage.rings.finite_rings 1
- lift_x(x, all=False, extend=False)#
Return one or all points with given \(x\)-coordinate.
This method is deterministic: It returns the same data each time when called again with the same \(x\).
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 : -1 : 1) sage: E.lift_x(2) (2 : -3 : 1) sage: E.lift_x(1/4, all=True) [(1/4 : -5/8 : 1), (1/4 : -3/8 : 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 # optional - sage.rings.number_field (3 : -y - 1 : 1) sage: P.curve() # optional - sage.rings.number_field 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 : -5.42442890089805 : 1.00000000000000), (3.00000000000000 : 4.42442890089805 : 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 # optional - sage.rings.number_field (0 : -y : 1) sage: P.curve() # optional - sage.rings.number_field 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 # optional - sage.rings.finite_rings Elliptic Curve defined by y^2 + y = x^3 + 16*x over Finite Field of size 17 sage: E.lift_x(7) # optional - sage.rings.finite_rings (7 : 5 : 1) sage: E.lift_x(3) # optional - sage.rings.finite_rings 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) # optional - sage.rings.finite_rings [(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 # optional - sage.rings.padics (6 + O(17^5) : 14 + O(17^5) : 1 + O(17^5)) sage: P.curve() # optional - sage.rings.padics 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 : -1 - 2*t + t^2 - 5*t^3 + 21*t^4 + O(t^5) : 1) sage: K.<a> = GF(16) # optional - sage.rings.finite_rings sage: P = E.change_ring(K).lift_x(a^3); P # optional - sage.rings.finite_rings (a^3 : a^3 + a : 1) sage: P.curve() # optional - sage.rings.finite_rings 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 github issue #30297 is fixed:
sage: K = Qp(5) # optional - sage.rings.padics sage: E = EllipticCurve([K(0), K(1)]) # optional - sage.rings.padics sage: E.lift_x(1, extend=True) # optional - sage.rings.padics (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') # optional - sage.rings.number_field sage: E.montgomery_model() # optional - sage.rings.number_field Elliptic Curve defined by y^2 = x^3 + (-1.953522420987248?)*x^2 + x over Algebraic Field
sage: E = EllipticCurve(GF(431^2), [7,7]) # optional - sage.rings.finite_rings sage: E.montgomery_model() # optional - sage.rings.finite_rings 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) # optional - sage.rings.finite_rings (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]) # optional - sage.rings.finite_rings sage: E.montgomery_model() # optional - sage.rings.finite_rings 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]) # optional - sage.rings.finite_rings sage: E.montgomery_model() # optional - sage.rings.finite_rings 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) # optional - sage.rings.finite_rings 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) # optional - sage.rings.finite_rings sage: f # optional - sage.rings.finite_rings 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 # optional - sage.rings.finite_rings 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) # optional - sage.rings.finite_rings sage: Q = C(17, 17) # optional - sage.rings.finite_rings sage: P + Q # this doesn't work... # optional - sage.rings.finite_rings Traceback (most recent call last): ... TypeError: unsupported operand parent(s) for +: ... sage: f(g(P) + g(Q)) # ...but this does # optional - sage.rings.finite_rings (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) # optional - sage.rings.finite_rings sage: C_ = C.change_ring(F) # optional - sage.rings.finite_rings sage: g_ = g.change_ring(F) # optional - sage.rings.finite_rings sage: g_(P).order() # optional - sage.rings.finite_rings 12 sage: T = C_(-7 * z2 - 57, 31 * z2 - 52, 1) # optional - sage.rings.finite_rings sage: g_(T).order() # optional - sage.rings.finite_rings 12 sage: g_(P).weil_pairing(g_(T), 12) # optional - sage.rings.finite_rings 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_ # optional - sage.rings.finite_rings 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 # optional - sage.rings.finite_rings -x^3 + 8*x^2*z - 127*y^2*z - x*z^2 sage: C_.is_isomorphic(EllipticCurve_from_cubic(h).codomain()) # optional - sage.rings.finite_rings 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
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 github issue #7368 and github issue #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) doctest:warning ... DeprecationWarning: ... 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() # optional - sage.libs.pari sage: type(e) # optional - sage.libs.pari <... 'cypari2.gen.Gen'> sage: e.type() # optional - sage.libs.pari 't_VEC' sage: e.disc() # optional - sage.libs.pari 37.0000000000000
Over a finite field:
sage: EllipticCurve(GF(41), [2,5]).pari_curve() # optional - sage.libs.pari sage.rings.finite_rings [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) # optional - sage.libs.pari sage.rings.padics sage: E = EllipticCurve(Qp, [3, 4]) # optional - sage.libs.pari sage.rings.padics sage: E.pari_curve() # optional - sage.libs.pari sage.rings.padics [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() # optional - sage.libs.pari sage.rings.padics 3*5^-1 + O(5)
Over a number field:
sage: K.<a> = QuadraticField(2) # optional - sage.libs.pari sage.rings.number_field sage: E = EllipticCurve([1,a]) # optional - sage.libs.pari sage.rings.number_field sage: E.pari_curve() # optional - sage.libs.pari sage.rings.number_field [0, 0, 0, Mod(1, y^2 - 2), Mod(y, y^2 - 2), 0, 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]) # optional - sage.libs.pari sage.rings.padics sage: E.j_invariant() # the j-invariant is a p-adic integer # optional - sage.libs.pari sage.rings.padics 2 + 4*5^2 + O(5^3) sage: E.pari_curve() # optional - sage.libs.pari sage.rings.padics [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)) # optional - sage.plot Graphics object consisting of 1 graphics primitive sage: E = EllipticCurve('37a') sage: plot(E) # optional - sage.plot Graphics object consisting of 2 graphics primitives sage: plot(E, xmin=25, xmax=26) # optional - sage.plot Graphics object consisting of 2 graphics primitives
With github issue #12766 we added the components keyword:
sage: E.real_components() 2 sage: E.plot(components='bounded') # optional - sage.plot Graphics object consisting of 1 graphics primitive sage: E.plot(components='unbounded') # optional - sage.plot 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') # optional - sage.plot 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() # optional - sage.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
- scalar_multiplication(m)#
Return the scalar-multiplication map \([m]\) on this elliptic curve as a
sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar
object.EXAMPLES:
sage: E = EllipticCurve('77a1') sage: m = E.scalar_multiplication(-7); m Scalar-multiplication endomorphism [-7] of Elliptic Curve defined by y^2 + y = x^3 + 2*x over Rational Field sage: m.degree() 49 sage: P = E(2,3) sage: m(P) (-26/225 : -2132/3375 : 1) sage: m.rational_maps() == E.multiplication_by_m(-7) True
- 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]) # optional - sage.rings.finite_rings sage: E.short_weierstrass_model(complete_cube=False) # optional - sage.rings.finite_rings Elliptic Curve defined by y^2 = x^3 + x + 2 over Finite Field of size 3
This used to be different see github issue #3973:
sage: E.short_weierstrass_model() # optional - sage.rings.finite_rings 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]) # optional - sage.rings.finite_rings sage: E.short_weierstrass_model() # optional - sage.rings.finite_rings 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) # optional - sage.rings.finite_rings 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) # optional - sage.rings.finite_rings 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]) # optional - sage.rings.finite_rings sage: E.two_division_polynomial() # optional - sage.rings.finite_rings x^3 + 2*x^2 + 2 sage: E.two_division_polynomial().roots() # optional - sage.rings.finite_rings [(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