Points on elliptic curves#

The base class EllipticCurvePoint_field, derived from AdditiveGroupElement, provides support for points on elliptic curves defined over general fields. The derived classes EllipticCurvePoint_number_field and EllipticCurvePoint_finite_field provide further support for point on curves defined over number fields (including the rational field \(\QQ\)) and over finite fields.

The class EllipticCurvePoint, which is based on SchemeMorphism_point_projective_ring, currently has little extra functionality.

EXAMPLES:

An example over \(\QQ\):

sage: E = EllipticCurve('389a1')
sage: P = E(-1,1); P
(-1 : 1 : 1)
sage: Q = E(0,-1); Q
(0 : -1 : 1)
sage: P+Q
(4 : 8 : 1)
sage: P-Q
(1 : 0 : 1)
sage: 3*P-5*Q
(328/361 : -2800/6859 : 1)

An example over a number field:

sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve(K,[1,0,0,0,-1])
sage: P = E(0,i); P
(0 : i : 1)
sage: P.order()
+Infinity
sage: 101*P-100*P == P
True

An example over a finite field:

sage: K.<a> = GF((101,3))
sage: E = EllipticCurve(K,[1,0,0,0,-1])
sage: P = E(40*a^2 + 69*a + 84 , 58*a^2 + 73*a + 45)
sage: P.order()
1032210
sage: E.cardinality()
1032210

Arithmetic with a point over an extension of a finite field:

sage: k.<a> = GF((5,2))
sage: E = EllipticCurve(k,[1,0]); E
Elliptic Curve defined by y^2 = x^3 + x over Finite Field in a of size 5^2
sage: P = E([a,2*a+4])
sage: 5*P
(2*a + 3 : 2*a : 1)
sage: P*5
(2*a + 3 : 2*a : 1)
sage: P + P + P + P + P
(2*a + 3 : 2*a : 1)

sage: F = Zmod(3)
sage: E = EllipticCurve(F,[1,0]);
sage: P = E([2,1])
sage: import sys
sage: n = sys.maxsize
sage: P*(n+1)-P*n == P
True

Arithmetic over \(\ZZ/N\ZZ\) with composite \(N\) is supported. When an operation tries to invert a non-invertible element, a ZeroDivisionError is raised and a factorization of the modulus appears in the error message:

sage: N = 1715761513
sage: E = EllipticCurve(Integers(N),[3,-13])
sage: P = E(2,1)
sage: LCM([2..60])*P
Traceback (most recent call last):
...
ZeroDivisionError: Inverse of 26927 does not exist (characteristic = 1715761513 = 26927*63719)

AUTHORS:

  • William Stein (2005) – Initial version

  • Robert Bradshaw et al….

  • John Cremona (Feb 2008) – Point counting and group structure for non-prime fields, Frobenius endomorphism and order, elliptic logs

  • John Cremona (Aug 2008) – Introduced EllipticCurvePoint_number_field class

  • Tobias Nagel, Michael Mardaus, John Cremona (Dec 2008) – \(p\)-adic elliptic logarithm over \(\QQ\)

  • David Hansen (Jan 2009) – Added weil_pairing function to EllipticCurvePoint_finite_field class

  • Mariah Lenox (March 2011) – Added tate_pairing and ate_pairing functions to EllipticCurvePoint_finite_field class

class sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint(X, v, check=True)#

Bases: SchemeMorphism_point_projective_ring

A point on an elliptic curve.

class sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_field(curve, v, check=True)#

Bases: SchemeMorphism_point_abelian_variety_field

A point on an elliptic curve over a field. The point has coordinates in the base field.

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: E([0,0])
(0 : 0 : 1)
sage: E(0,0)               # brackets are optional
(0 : 0 : 1)
sage: E([GF(5)(0), 0])     # entries are coerced
(0 : 0 : 1)

sage: E(0.000, 0)
(0 : 0 : 1)

sage: E(1,0,0)
Traceback (most recent call last):
...
TypeError: Coordinates [1, 0, 0] do not define a point on
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field

sage: E = EllipticCurve([0,0,1,-1,0])
sage: S = E(QQ); S
Abelian group of points on Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field

sage: K.<i>=NumberField(x^2+1)
sage: E=EllipticCurve(K,[0,1,0,-160,308])
sage: P=E(26,-120)
sage: Q=E(2+12*i,-36+48*i)
sage: P.order() == Q.order() == 4  # long time (3s)
True
sage: 2*P==2*Q
False

sage: K.<t>=FractionField(PolynomialRing(QQ,'t'))
sage: E=EllipticCurve([0,0,0,0,t^2])
sage: P=E(0,t)
sage: P,2*P,3*P
((0 : t : 1), (0 : -t : 1), (0 : 1 : 0))
additive_order()#

Return the order of this point on the elliptic curve.

If the point is zero, returns 1, otherwise raise a NotImplementedError.

For curves over number fields and finite fields, see below.

Note

additive_order() is a synonym for order()

EXAMPLES:

sage: K.<t>=FractionField(PolynomialRing(QQ,'t'))
sage: E=EllipticCurve([0,0,0,-t^2,0])
sage: P=E(t,0)
sage: P.order()
Traceback (most recent call last):
...
NotImplementedError: Computation of order of a point not implemented over general fields.
sage: E(0).additive_order()
1
sage: E(0).order() == 1
True
ate_pairing(Q, n, k, t, q=None)#

Return ate pairing of \(n\)-torsion points \(P=self\) and \(Q\).

Also known as the \(n\)-th modified ate pairing. \(P\) is \(GF(q)\)-rational, and \(Q\) must be an element of \(Ker(\pi-p)\), where \(\pi\) is the \(q\)-frobenius map (and hence \(Q\) is \(GF(q^k)\)-rational).

INPUT:

  • P=self – a point of order \(n\), in \(ker(\pi-1)\), where \(\pi\) is the \(q\)-Frobenius map (e.g., \(P\) is \(q-rational\)).

  • Q – a point of order \(n\) in \(ker(\pi-q)\)

  • n – the order of \(P\) and \(Q\).

  • k – the embedding degree.

  • t – the trace of Frobenius of the curve over \(GF(q)\).

  • q – (default: None) the size of base field (the “big” field is \(GF(q^k)\)). \(q\) needs to be set only if its value cannot be deduced.

OUTPUT:

FiniteFieldElement in \(GF(q^k)\) – the ate pairing of \(P\) and \(Q\).

EXAMPLES:

An example with embedding degree 6:

sage: p = 7549; A = 0; B = 1; n = 157; k = 6; t = 14
sage: F = GF(p); E = EllipticCurve(F, [A, B])
sage: R.<x> = F[]; K.<a> = GF((p,k), modulus=x^k+2)
sage: EK = E.base_extend(K)
sage: P = EK(3050, 5371); Q = EK(6908*a^4, 3231*a^3)
sage: P.ate_pairing(Q, n, k, t)
6708*a^5 + 4230*a^4 + 4350*a^3 + 2064*a^2 + 4022*a + 6733
sage: s = Integer(randrange(1, n))
sage: (s*P).ate_pairing(Q, n, k, t) == P.ate_pairing(s*Q, n, k, t)
True
sage: P.ate_pairing(s*Q, n, k, t) == P.ate_pairing(Q, n, k, t)^s
True

Another example with embedding degree 7 and positive trace:

sage: p = 2213; A = 1; B = 49; n = 1093; k = 7; t = 28
sage: F = GF(p); E = EllipticCurve(F, [A, B])
sage: R.<x> = F[]; K.<a> = GF((p,k), modulus=x^k+2)
sage: EK = E.base_extend(K)
sage: P = EK(1583, 1734)
sage: Qx = 1729*a^6+1767*a^5+245*a^4+980*a^3+1592*a^2+1883*a+722
sage: Qy = 1299*a^6+1877*a^5+1030*a^4+1513*a^3+1457*a^2+309*a+1636
sage: Q = EK(Qx, Qy)
sage: P.ate_pairing(Q, n, k, t)
1665*a^6 + 1538*a^5 + 1979*a^4 + 239*a^3 + 2134*a^2 + 2151*a + 654
sage: s = Integer(randrange(1, n))
sage: (s*P).ate_pairing(Q, n, k, t) == P.ate_pairing(s*Q, n, k, t)
True
sage: P.ate_pairing(s*Q, n, k, t) == P.ate_pairing(Q, n, k, t)^s
True

Another example with embedding degree 7 and negative trace:

sage: p = 2017; A = 1; B = 30; n = 29; k = 7; t = -70
sage: F = GF(p); E = EllipticCurve(F, [A, B])
sage: R.<x> = F[]; K.<a> = GF((p,k), modulus=x^k+2)
sage: EK = E.base_extend(K)
sage: P = EK(369, 716)
sage: Qx = 1226*a^6+1778*a^5+660*a^4+1791*a^3+1750*a^2+867*a+770
sage: Qy = 1764*a^6+198*a^5+1206*a^4+406*a^3+1200*a^2+273*a+1712
sage: Q = EK(Qx, Qy)
sage: P.ate_pairing(Q, n, k, t)
1794*a^6 + 1161*a^5 + 576*a^4 + 488*a^3 + 1950*a^2 + 1905*a + 1315
sage: s = Integer(randrange(1, n))
sage: (s*P).ate_pairing(Q, n, k, t) == P.ate_pairing(s*Q, n, k, t)
True
sage: P.ate_pairing(s*Q, n, k, t) == P.ate_pairing(Q, n, k, t)^s
True

Using the same data, we show that the ate pairing is a power of the Tate pairing (see [HSV2006] end of section 3.1):

sage: c = (k*p^(k-1)).mod(n); T = t - 1
sage: N = gcd(T^k - 1, p^k - 1)
sage: s = Integer(N/n)
sage: L = Integer((T^k - 1)/N)
sage: M = (L*s*c.inverse_mod(n)).mod(n)
sage: P.ate_pairing(Q, n, k, t) == Q.tate_pairing(P, n, k)^M
True

An example where we have to pass the base field size (and we again have agreement with the Tate pairing). Note that though \(Px\) is not \(F\)-rational, (it is the homomorphic image of an \(F\)-rational point) it is nonetheless in \(ker(\pi-1)\), and so is a legitimate input:

sage: q = 2^5; F.<a>=GF(q)
sage: n = 41; k = 4; t = -8
sage: E=EllipticCurve(F,[0,0,1,1,1])
sage: P = E(a^4 + 1, a^3)
sage: Fx.<b>=GF(q^k)
sage: Ex=EllipticCurve(Fx,[0,0,1,1,1])
sage: phi=Hom(F,Fx)(F.gen().minpoly().roots(Fx)[0][0])
sage: Px=Ex(phi(P.xy()[0]),phi(P.xy()[1]))
sage: Qx = Ex(b^19+b^18+b^16+b^12+b^10+b^9+b^8+b^5+b^3+1, b^18+b^13+b^10+b^8+b^5+b^4+b^3+b)
sage: Qx = Ex(Qx[0]^q, Qx[1]^q) - Qx  # ensure Qx is in ker(pi - q)
sage: Px.ate_pairing(Qx, n, k, t)
Traceback (most recent call last):
...
ValueError: Unexpected field degree: set keyword argument q equal to the size of the base field (big field is GF(q^4)).
sage: Px.ate_pairing(Qx, n, k, t, q)
b^19 + b^18 + b^17 + b^16 + b^15 + b^14 + b^13 + b^12 + b^11 + b^9 + b^8 + b^5 + b^4 + b^2 + b + 1
sage: s = Integer(randrange(1, n))
sage: (s*Px).ate_pairing(Qx, n, k, t, q) == Px.ate_pairing(s*Qx, n, k, t, q)
True
sage: Px.ate_pairing(s*Qx, n, k, t, q) == Px.ate_pairing(Qx, n, k, t, q)^s
True
sage: c = (k*q^(k-1)).mod(n); T = t - 1
sage: N = gcd(T^k - 1, q^k - 1)
sage: s = Integer(N/n)
sage: L = Integer((T^k - 1)/N)
sage: M = (L*s*c.inverse_mod(n)).mod(n)
sage: Px.ate_pairing(Qx, n, k, t, q) == Qx.tate_pairing(Px, n, k, q)^M
True

It is an error if \(Q\) is not in the kernel of \(\pi - p\), where \(\pi\) is the Frobenius automorphism:

sage: p = 29; A = 1; B = 0; n = 5; k = 2; t = 10
sage: F = GF(p); R.<x> = F[]
sage: E = EllipticCurve(F, [A, B]);
sage: K.<a> = GF((p,k), modulus=x^k+2); EK = E.base_extend(K)
sage: P = EK(13, 8); Q = EK(13, 21)
sage: P.ate_pairing(Q, n, k, t)
Traceback (most recent call last):
...
ValueError: Point (13 : 21 : 1) not in Ker(pi - q)

It is also an error if \(P\) is not in the kernel os \(\pi - 1\):

sage: p = 29; A = 1; B = 0; n = 5; k = 2; t = 10
sage: F = GF(p); R.<x> = F[]
sage: E = EllipticCurve(F, [A, B]);
sage: K.<a> = GF((p,k), modulus=x^k+2); EK = E.base_extend(K)
sage: P = EK(14, 10*a); Q = EK(13, 21)
sage: P.ate_pairing(Q, n, k, t)
Traceback (most recent call last):
...
ValueError: This point (14 : 10*a : 1) is not in Ker(pi - 1)

Note

First defined in the paper of [HSV2006], the ate pairing can be computationally effective in those cases when the trace of the curve over the base field is significantly smaller than the expected value. This implementation is simply Miller’s algorithm followed by a naive exponentiation, and makes no claims towards efficiency.

AUTHORS:

  • Mariah Lenox (2011-03-08)

curve()#

Return the curve that this point is on.

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: P = E([-1,1])
sage: P.curve()
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
division_points(m, poly_only=False)#

Return a list of all points \(Q\) such that \(mQ=P\) where \(P\) = self.

Only points on the elliptic curve containing self and defined over the base field are included.

INPUT:

  • m – a positive integer

  • poly_only – bool (default: False); if True return polynomial whose roots give all possible \(x\)-coordinates of \(m\)-th roots of self.

OUTPUT:

(list) – a (possibly empty) list of solutions \(Q\) to \(mQ=P\), where \(P\) = self.

EXAMPLES:

We find the five 5-torsion points on an elliptic curve:

sage: E = EllipticCurve('11a'); E
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sage: P = E(0); P
(0 : 1 : 0)
sage: P.division_points(5)
[(0 : 1 : 0), (5 : -6 : 1), (5 : 5 : 1), (16 : -61 : 1), (16 : 60 : 1)]

Note above that 0 is included since [5]*0 = 0.

We create a curve of rank 1 with no torsion and do a consistency check:

sage: E = EllipticCurve('11a').quadratic_twist(-7)
sage: Q = E([44,-270])
sage: (4*Q).division_points(4)
[(44 : -270 : 1)]

We create a curve over a non-prime finite field with group of order \(18\):

sage: k.<a> = GF((5,2))
sage: E = EllipticCurve(k, [1,2+a,3,4*a,2])
sage: P = E([3,3*a+4])
sage: factor(E.order())
2 * 3^2
sage: P.order()
9

We find the \(1\)-division points as a consistency check – there is just one, of course:

sage: P.division_points(1)
[(3 : 3*a + 4 : 1)]

The point \(P\) has order coprime to 2 but divisible by 3, so:

sage: P.division_points(2)
[(2*a + 1 : 3*a + 4 : 1), (3*a + 1 : a : 1)]

We check that each of the 2-division points works as claimed:

sage: [2*Q for Q in P.division_points(2)]
[(3 : 3*a + 4 : 1), (3 : 3*a + 4 : 1)]

Some other checks:

sage: P.division_points(3)
[]
sage: P.division_points(4)
[(0 : 3*a + 2 : 1), (1 : 0 : 1)]
sage: P.division_points(5)
[(1 : 1 : 1)]

An example over a number field (see trac ticket #3383):

sage: E = EllipticCurve('19a1')
sage: K.<t> = NumberField(x^9-3*x^8-4*x^7+16*x^6-3*x^5-21*x^4+5*x^3+7*x^2-7*x+1)
sage: EK = E.base_extend(K)
sage: E(0).division_points(3)
[(0 : 1 : 0), (5 : -10 : 1), (5 : 9 : 1)]
sage: EK(0).division_points(3)
[(0 : 1 : 0), (5 : 9 : 1), (5 : -10 : 1)]
sage: E(0).division_points(9)
[(0 : 1 : 0), (5 : -10 : 1), (5 : 9 : 1)]
sage: EK(0).division_points(9)
[(0 : 1 : 0), (5 : 9 : 1), (5 : -10 : 1), (-150/121*t^8 + 414/121*t^7 + 1481/242*t^6 - 2382/121*t^5 - 103/242*t^4 + 629/22*t^3 - 367/242*t^2 - 1307/121*t + 625/121 : 35/484*t^8 - 133/242*t^7 + 445/242*t^6 - 799/242*t^5 + 373/484*t^4 + 113/22*t^3 - 2355/484*t^2 - 753/242*t + 1165/484 : 1), (-150/121*t^8 + 414/121*t^7 + 1481/242*t^6 - 2382/121*t^5 - 103/242*t^4 + 629/22*t^3 - 367/242*t^2 - 1307/121*t + 625/121 : -35/484*t^8 + 133/242*t^7 - 445/242*t^6 + 799/242*t^5 - 373/484*t^4 - 113/22*t^3 + 2355/484*t^2 + 753/242*t - 1649/484 : 1), (-1383/484*t^8 + 970/121*t^7 + 3159/242*t^6 - 5211/121*t^5 + 37/484*t^4 + 654/11*t^3 - 909/484*t^2 - 4831/242*t + 6791/484 : 927/121*t^8 - 5209/242*t^7 - 8187/242*t^6 + 27975/242*t^5 - 1147/242*t^4 - 1729/11*t^3 + 1566/121*t^2 + 12873/242*t - 10871/242 : 1), (-1383/484*t^8 + 970/121*t^7 + 3159/242*t^6 - 5211/121*t^5 + 37/484*t^4 + 654/11*t^3 - 909/484*t^2 - 4831/242*t + 6791/484 : -927/121*t^8 + 5209/242*t^7 + 8187/242*t^6 - 27975/242*t^5 + 1147/242*t^4 + 1729/11*t^3 - 1566/121*t^2 - 12873/242*t + 10629/242 : 1), (-4793/484*t^8 + 6791/242*t^7 + 10727/242*t^6 - 18301/121*t^5 + 2347/484*t^4 + 2293/11*t^3 - 7311/484*t^2 - 17239/242*t + 26767/484 : 30847/484*t^8 - 21789/121*t^7 - 34605/121*t^6 + 117164/121*t^5 - 10633/484*t^4 - 29437/22*t^3 + 39725/484*t^2 + 55428/121*t - 176909/484 : 1), (-4793/484*t^8 + 6791/242*t^7 + 10727/242*t^6 - 18301/121*t^5 + 2347/484*t^4 + 2293/11*t^3 - 7311/484*t^2 - 17239/242*t + 26767/484 : -30847/484*t^8 + 21789/121*t^7 + 34605/121*t^6 - 117164/121*t^5 + 10633/484*t^4 + 29437/22*t^3 - 39725/484*t^2 - 55428/121*t + 176425/484 : 1)]
has_finite_order()#

Return True if this point has finite additive order as an element of the group of points on this curve.

For fields other than number fields and finite fields, this is NotImplemented unless self.is_zero().

EXAMPLES:

sage: K.<t>=FractionField(PolynomialRing(QQ,'t'))
sage: E=EllipticCurve([0,0,0,-t^2,0])
sage: P = E(0)
sage: P.has_finite_order()
True
sage: P=E(t,0)
sage: P.has_finite_order()
Traceback (most recent call last):
...
NotImplementedError: Computation of order of a point not implemented over general fields.
sage: (2*P).is_zero()
True
has_infinite_order()#

Return True if this point has infinite additive order as an element of the group of points on this curve.

For fields other than number fields and finite fields, this is NotImplemented unless self.is_zero().

EXAMPLES:

sage: K.<t>=FractionField(PolynomialRing(QQ,'t'))
sage: E=EllipticCurve([0,0,0,-t^2,0])
sage: P = E(0)
sage: P.has_infinite_order()
False
sage: P=E(t,0)
sage: P.has_infinite_order()
Traceback (most recent call last):
...
NotImplementedError: Computation of order of a point not implemented over general fields.
sage: (2*P).is_zero()
True
is_divisible_by(m)#

Return True if there exists a point \(Q\) defined over the same field as self such that \(mQ\) == self.

INPUT:

  • m – a positive integer.

OUTPUT:

(bool) – True if there is a solution, else False.

Warning

This function usually triggers the computation of the \(m\)-th division polynomial of the associated elliptic curve, which will be expensive if \(m\) is large, though it will be cached for subsequent calls with the same \(m\).

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: Q = 5*E(0,0); Q
(-2739/1444 : -77033/54872 : 1)
sage: Q.is_divisible_by(4)
False
sage: Q.is_divisible_by(5)
True

A finite field example:

sage: E = EllipticCurve(GF(101),[23,34])
sage: E.cardinality().factor()
2 * 53
sage: Set([T.order() for T in E.points()])
{1, 106, 2, 53}
sage: len([T for T in E.points() if T.is_divisible_by(2)])
53
sage: len([T for T in E.points() if T.is_divisible_by(3)])
106
is_finite_order()#

Return True if this point has finite additive order as an element of the group of points on this curve.

For fields other than number fields and finite fields, this is NotImplemented unless self.is_zero().

EXAMPLES:

sage: K.<t>=FractionField(PolynomialRing(QQ,'t'))
sage: E=EllipticCurve([0,0,0,-t^2,0])
sage: P = E(0)
sage: P.has_finite_order()
True
sage: P=E(t,0)
sage: P.has_finite_order()
Traceback (most recent call last):
...
NotImplementedError: Computation of order of a point not implemented over general fields.
sage: (2*P).is_zero()
True
order()#

Return the order of this point on the elliptic curve.

If the point is zero, returns 1, otherwise raise a NotImplementedError.

For curves over number fields and finite fields, see below.

Note

additive_order() is a synonym for order()

EXAMPLES:

sage: K.<t>=FractionField(PolynomialRing(QQ,'t'))
sage: E=EllipticCurve([0,0,0,-t^2,0])
sage: P=E(t,0)
sage: P.order()
Traceback (most recent call last):
...
NotImplementedError: Computation of order of a point not implemented over general fields.
sage: E(0).additive_order()
1
sage: E(0).order() == 1
True
plot(**args)#

Plot this point on an elliptic curve.

INPUT:

  • **args – all arguments get passed directly onto the point plotting function.

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: P = E([-1,1])
sage: P.plot(pointsize=30, rgbcolor=(1,0,0))
Graphics object consisting of 1 graphics primitive
scheme()#

Return the scheme of this point, i.e., the curve it is on. This is synonymous with curve() which is perhaps more intuitive.

EXAMPLES:

sage: E=EllipticCurve(QQ,[1,1])
sage: P=E(0,1)
sage: P.scheme()
Elliptic Curve defined by y^2 = x^3 + x + 1 over Rational Field
sage: P.scheme() == P.curve()
True
sage: K.<a>=NumberField(x^2-3,'a')
sage: P=E.base_extend(K)(1,a)
sage: P.scheme()
Elliptic Curve defined by y^2 = x^3 + x + 1 over Number Field in a with defining polynomial x^2 - 3
set_order(value, check)#

Set the value of self._order to value.

Use this when you know a priori the order of this point to avoid a potentially expensive order calculation.

INPUT:

  • value – positive integer

OUTPUT:

None

EXAMPLES:

This example illustrates basic usage.

sage: E = EllipticCurve(GF(7), [0, 1])  # This curve has order 12
sage: G = E(5, 0)
sage: G.set_order(2)
sage: 2*G
(0 : 1 : 0)

We now give a more interesting case, the NIST-P521 curve. Its order is too big to calculate with Sage, and takes a long time using other packages, so it is very useful here.

sage: p = 2^521 - 1
sage: prev_proof_state = proof.arithmetic()
sage: proof.arithmetic(False) # turn off primality checking
sage: F = GF(p)
sage: A = p - 3
sage: B = 1093849038073734274511112390766805569936207598951683748994586394495953116150735016013708737573759623248592132296706313309438452531591012912142327488478985984
sage: q = 6864797660130609714981900799081393217269435300143305409394463459185543183397655394245057746333217197532963996371363321113864768612440380340372808892707005449
sage: E = EllipticCurve([F(A), F(B)])
sage: G = E.random_point()
sage: G.set_order(q)
sage: G.order() * G  # This takes practically no time.
(0 : 1 : 0)
sage: proof.arithmetic(prev_proof_state) # restore state

It is an error to pass a \(value\) equal to \(0\):

sage: E = EllipticCurve(GF(7), [0, 1])  # This curve has order 12
sage: G = E.random_point()
sage: G.set_order(0)
Traceback (most recent call last):
...
ValueError: Value 0 illegal for point order
sage: G.set_order(1000)
Traceback (most recent call last):
...
ValueError: Value 1000 illegal: outside max Hasse bound

It is also very likely an error to pass a value which is not the actual order of this point. How unlikely is determined by the factorization of the actual order, and the actual group structure:

sage: E = EllipticCurve(GF(7), [0, 1])  # This curve has order 12
sage: G = E(5, 0)   # G has order 2
sage: G.set_order(11)
Traceback (most recent call last):
...
ValueError: Value 11 illegal: 11 * (5 : 0 : 1) is not the identity

However, set_order can be fooled, though it’s not likely in “real cases of interest”. For instance, the order can be set to a multiple the actual order:

sage: E = EllipticCurve(GF(7), [0, 1])  # This curve has order 12
sage: G = E(5, 0)   # G has order 2
sage: G.set_order(8)
sage: G.order()
8

AUTHORS:

  • Mariah Lenox (2011-02-16)

tate_pairing(Q, n, k, q=None)#

Return Tate pairing of \(n\)-torsion point \(P = self\) and point \(Q\).

The value returned is \(f_{n,P}(Q)^e\) where \(f_{n,P}\) is a function with divisor \(n[P]-n[O].\). This is also known as the “modified Tate pairing”. It is a well-defined bilinear map.

INPUT:

  • P=self – Elliptic curve point having order n

  • Q – Elliptic curve point on same curve as P (can be any order)

  • n – positive integer: order of P

  • k – positive integer: embedding degree

  • q – positive integer: size of base field (the “big” field is \(GF(q^k)\). \(q\) needs to be set only if its value cannot be deduced.)

OUTPUT:

An \(n\)’th root of unity in the base field self.curve().base_field()

EXAMPLES:

A simple example, pairing a point with itself, and pairing a point with another rational point:

sage: p = 103; A = 1; B = 18; E = EllipticCurve(GF(p), [A, B])
sage: P = E(33, 91); n = P.order(); n
19
sage: k = GF(n)(p).multiplicative_order(); k
6
sage: P.tate_pairing(P, n, k)
1
sage: Q = E(87, 51)
sage: P.tate_pairing(Q, n, k)
1
sage: set_random_seed(35)
sage: P.tate_pairing(P,n,k)
1

We now let Q be a point on the same curve as above, but defined over the pairing extension field, and we also demonstrate the bilinearity of the pairing:

sage: K.<a> = GF((p,k))
sage: EK = E.base_extend(K); P = EK(P)
sage: Qx = 69*a^5 + 96*a^4 + 22*a^3 + 86*a^2 + 6*a + 35
sage: Qy = 34*a^5 + 24*a^4 + 16*a^3 + 41*a^2 + 4*a + 40
sage: Q = EK(Qx, Qy);

Multiply by cofactor so Q has order n:

sage: h = 551269674; Q = h*Q
sage: P = EK(P); P.tate_pairing(Q, n, k)
24*a^5 + 34*a^4 + 3*a^3 + 69*a^2 + 86*a + 45
sage: s = Integer(randrange(1,n))
sage: ans1 = (s*P).tate_pairing(Q, n, k)
sage: ans2 = P.tate_pairing(s*Q, n, k)
sage: ans3 = P.tate_pairing(Q, n, k)^s
sage: ans1 == ans2 == ans3
True
sage: (ans1 != 1) and (ans1^n == 1)
True

Here is an example of using the Tate pairing to compute the Weil pairing (using the same data as above):

sage: e = Integer((p^k-1)/n); e
62844857712
sage: P.weil_pairing(Q, n)^e
94*a^5 + 99*a^4 + 29*a^3 + 45*a^2 + 57*a + 34
sage: P.tate_pairing(Q, n, k) == P._miller_(Q, n)^e
True
sage: Q.tate_pairing(P, n, k) == Q._miller_(P, n)^e
True
sage: P.tate_pairing(Q, n, k)/Q.tate_pairing(P, n, k)
94*a^5 + 99*a^4 + 29*a^3 + 45*a^2 + 57*a + 34

An example where we have to pass the base field size (and we again have agreement with the Weil pairing):

sage: F.<a>=GF((2,5))
sage: E=EllipticCurve(F,[0,0,1,1,1])
sage: P = E(a^4 + 1, a^3)
sage: Fx.<b>=GF((2,4*5))
sage: Ex=EllipticCurve(Fx,[0,0,1,1,1])
sage: phi=Hom(F,Fx)(F.gen().minpoly().roots(Fx)[0][0])
sage: Px=Ex(phi(P.xy()[0]),phi(P.xy()[1]))
sage: Qx = Ex(b^19+b^18+b^16+b^12+b^10+b^9+b^8+b^5+b^3+1, b^18+b^13+b^10+b^8+b^5+b^4+b^3+b)
sage: Px.tate_pairing(Qx, n=41, k=4)
Traceback (most recent call last):
...
ValueError: Unexpected field degree: set keyword argument q equal to the size of the base field (big field is GF(q^4)).
sage: num = Px.tate_pairing(Qx, n=41, k=4, q=32); num
b^19 + b^14 + b^13 + b^12 + b^6 + b^4 + b^3
sage: den = Qx.tate_pairing(Px, n=41, k=4, q=32); den
b^19 + b^17 + b^16 + b^15 + b^14 + b^10 + b^6 + b^2 + 1
sage: e = Integer((32^4-1)/41); e
25575
sage: Px.weil_pairing(Qx, 41)^e == num/den
True

Note

This function uses Miller’s algorithm, followed by a naive exponentiation. It does not do anything fancy. In the case that there is an issue with \(Q\) being on one of the lines generated in the \(r*P\) calculation, \(Q\) is offset by a random point \(R\) and P.tate_pairing(Q+R,n,k)/P.tate_pairing(R,n,k) is returned.

AUTHORS:

  • Mariah Lenox (2011-03-07)

weil_pairing(Q, n, algorithm=None)#

Compute the Weil pairing of this point with another point \(Q\) on the same curve.

INPUT:

  • Q – another point on the same curve as self.

  • n – an integer \(n\) such that \(nP = nQ = (0:1:0)\), where \(P\) is self.

  • algorithm (default: None) – choices are pari and sage. PARI is usually significantly faster, but it only works over finite fields. When None is given, a suitable algorithm is chosen automatically.

OUTPUT:

An \(n\)’th root of unity in the base field of the curve.

EXAMPLES:

sage: F.<a>=GF((2,5))
sage: E=EllipticCurve(F,[0,0,1,1,1])
sage: P = E(a^4 + 1, a^3)
sage: Fx.<b>=GF((2,4*5))
sage: Ex=EllipticCurve(Fx,[0,0,1,1,1])
sage: phi=Hom(F,Fx)(F.gen().minpoly().roots(Fx)[0][0])
sage: Px=Ex(phi(P.xy()[0]),phi(P.xy()[1]))
sage: O = Ex(0)
sage: Qx = Ex(b^19 + b^18 + b^16 + b^12 + b^10 + b^9 + b^8 + b^5 + b^3 + 1, b^18 + b^13 + b^10 + b^8 + b^5 + b^4 + b^3 + b)
sage: Px.weil_pairing(Qx,41) == b^19 + b^15 + b^9 + b^8 + b^6 + b^4 + b^3 + b^2 + 1
True
sage: Px.weil_pairing(17*Px,41) == Fx(1)
True
sage: Px.weil_pairing(O,41) == Fx(1)
True

An error is raised if either point is not \(n\)-torsion:

sage: Px.weil_pairing(O,40)
Traceback (most recent call last):
...
ValueError: points must both be n-torsion

A larger example (see trac ticket #4964):

sage: P,Q = EllipticCurve(GF((19,4),'a'),[-1,0]).gens()
sage: P.order(), Q.order()
(360, 360)
sage: z = P.weil_pairing(Q,360)
sage: z.multiplicative_order()
360

An example over a number field:

sage: P,Q = EllipticCurve('11a1').change_ring(CyclotomicField(5)).torsion_subgroup().gens()
sage: P,Q = (P.element(), Q.element())
sage: (P.order(),Q.order())
(5, 5)
sage: P.weil_pairing(Q,5)
zeta5^2
sage: Q.weil_pairing(P,5)
zeta5^3

ALGORITHM:

  • For algorithm='pari': pari:ellweilpairing.

  • For algorithm='sage': Implemented using Proposition 8 in [Mil2004]. The value 1 is returned for linearly dependent input points. This condition is caught via a DivisionByZeroError, since the use of a discrete logarithm test for linear dependence is much too slow for large \(n\).

AUTHORS:

  • David Hansen (2009-01-25)

  • Lorenz Panny (2022): algorithm='pari'

xy()#

Return the \(x\) and \(y\) coordinates of this point, as a 2-tuple. If this is the point at infinity a ZeroDivisionError is raised.

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: P = E([-1,1])
sage: P.xy()
(-1, 1)
sage: Q = E(0); Q
(0 : 1 : 0)
sage: Q.xy()
Traceback (most recent call last):
...
ZeroDivisionError: rational division by zero
class sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_finite_field(curve, v, check=True)#

Bases: EllipticCurvePoint_field

Class for elliptic curve points over finite fields.

additive_order()#

Return the order of this point on the elliptic curve.

ALGORITHM: Use PARI function pari:ellorder.

Note

additive_order() is a synonym for order()

EXAMPLES:

sage: k.<a> = GF((5,5))
sage: E = EllipticCurve(k,[2,4]); E
Elliptic Curve defined by y^2 = x^3 + 2*x + 4 over Finite Field in a of size 5^5
sage: P = E(3*a^4 + 3*a , 2*a + 1 )
sage: P.order()
3227
sage: Q = E(0,2)
sage: Q.order()
7
sage: Q.additive_order()
7

sage: p=next_prime(2^150)
sage: E=EllipticCurve(GF(p),[1,1])
sage: P=E(831623307675610677632782670796608848711856078, 42295786042873366706573292533588638217232964)
sage: P.order()
1427247692705959881058262545272474300628281448
sage: P.order() == E.cardinality()
True

The next example has \(j(E)=0\):

sage: p = 33554501
sage: F.<u> = GF((p,2))
sage: E = EllipticCurve(F,[0,1])
sage: E.j_invariant()
0
sage: P = E.random_point()
sage: P.order() # random
16777251

Similarly when \(j(E)=1728\):

sage: p = 33554473
sage: F.<u> = GF((p,2))
sage: E = EllipticCurve(F,[1,0])
sage: E.j_invariant()
1728
sage: P = E.random_point()
sage: P.order() # random
46912611635760
discrete_log(Q, ord=None)#

Return the discrete logarithm of \(Q\) to base \(P\) = self, that is, an integer \(x\) such that \(xP = Q\).

A ValueError is raised if there is no solution.

ALGORITHM:

To compute the actual logarithm, pari:elllog is called.

However, elllog() does not guarantee termination if \(Q\) is not a multiple of \(P\), so we first need to check subgroup membership. This is done as follows:

  • Let \(n\) denote the order of \(P\). First check that \(nQ\) equals the point at infinity (and hence the order of \(Q\) divides \(n\)).

  • If the curve order \(\#E\) has been cached, check whether \(\gcd(n^2, \#E) = n\). If this holds, the curve has cyclic \(n\)-torsion, hence all points whose order divides \(n\) must be multiples of \(P\) and we are done.

  • Otherwise (if this test is inconclusive), check that the Weil pairing of \(P\) and \(Q\) is trivial.

For anomalous curves with \(\#E = p\), the padic_elliptic_logarithm() function is called.

INPUT:

  • Q (point) – another point on the same curve as self.

OUTPUT:

(integer) – The discrete logarithm of \(Q\) with respect to \(P\), which is an integer \(x\) with \(0\le x<\mathrm{ord}(P)\) such that \(xP=Q\), if one exists.

AUTHORS:

  • John Cremona. Adapted to use generic functions 2008-04-05.

  • Lorenz Panny (2022): switch to PARI.

EXAMPLES:

sage: F = GF((3,6),'a')
sage: a = F.gen()
sage: E = EllipticCurve([0,1,1,a,a])
sage: E.cardinality()
762
sage: P = E.gens()[0]
sage: Q = 400*P
sage: P.discrete_log(Q)
400
has_finite_order()#

Return True if this point has finite additive order as an element of the group of points on this curve.

Since the base field is finite, the answer will always be True.

EXAMPLES:

sage: E = EllipticCurve(GF(7), [1,3])
sage: P = E.points()[3]
sage: P.has_finite_order()
True
order()#

Return the order of this point on the elliptic curve.

ALGORITHM: Use PARI function pari:ellorder.

Note

additive_order() is a synonym for order()

EXAMPLES:

sage: k.<a> = GF((5,5))
sage: E = EllipticCurve(k,[2,4]); E
Elliptic Curve defined by y^2 = x^3 + 2*x + 4 over Finite Field in a of size 5^5
sage: P = E(3*a^4 + 3*a , 2*a + 1 )
sage: P.order()
3227
sage: Q = E(0,2)
sage: Q.order()
7
sage: Q.additive_order()
7

sage: p=next_prime(2^150)
sage: E=EllipticCurve(GF(p),[1,1])
sage: P=E(831623307675610677632782670796608848711856078, 42295786042873366706573292533588638217232964)
sage: P.order()
1427247692705959881058262545272474300628281448
sage: P.order() == E.cardinality()
True

The next example has \(j(E)=0\):

sage: p = 33554501
sage: F.<u> = GF((p,2))
sage: E = EllipticCurve(F,[0,1])
sage: E.j_invariant()
0
sage: P = E.random_point()
sage: P.order() # random
16777251

Similarly when \(j(E)=1728\):

sage: p = 33554473
sage: F.<u> = GF((p,2))
sage: E = EllipticCurve(F,[1,0])
sage: E.j_invariant()
1728
sage: P = E.random_point()
sage: P.order() # random
46912611635760
padic_elliptic_logarithm(Q, p)#

Return the discrete logarithm of \(Q\) to base \(P\) = self, that is, an integer \(x\) such that \(xP = Q\) only for anomalous curves.

ALGORITHM:

Discrete logarithm computed as in [Sma1999] with a loop to avoid the canonical lift.

INPUT:

  • Q (point) – another point on the same curve as self.

  • p (integer) – a prime equals the order of the curve.

OUTPUT:

(integer) – The discrete logarithm of \(Q\) with respect to \(P\), which is an integer \(x\) with \(0\le x<\mathrm{ord}(P)\) such that \(xP=Q\).

AUTHORS:

  • Sylvain Pelissier (2022) based on Samuel Neves code.

EXAMPLES:

sage: p=235322474717419
sage: b=8856682
sage: E = EllipticCurve(GF(p), [0, b])
sage: P = E(200673830421813, 57025307876612)
sage: Q = E(40345734829479, 211738132651297)
sage: x = P.padic_elliptic_logarithm(Q, p)
sage: x * P == Q
True
class sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_number_field(curve, v, check=True)#

Bases: EllipticCurvePoint_field

A point on an elliptic curve over a number field.

Most of the functionality is derived from the parent class EllipticCurvePoint_field. In addition we have support for orders, heights, reduction modulo primes, and elliptic logarithms.

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: E([0,0])
(0 : 0 : 1)
sage: E(0,0)               # brackets are optional
(0 : 0 : 1)
sage: E([GF(5)(0), 0])     # entries are coerced
(0 : 0 : 1)

sage: E(0.000, 0)
(0 : 0 : 1)

sage: E(1,0,0)
Traceback (most recent call last):
...
TypeError: Coordinates [1, 0, 0] do not define a point on
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field

sage: E = EllipticCurve([0,0,1,-1,0])
sage: S = E(QQ); S
Abelian group of points on Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
additive_order()#

Return the order of this point on the elliptic curve.

If the point has infinite order, returns +Infinity. For curves defined over \(\QQ\), we call PARI; over other number fields we implement the function here.

Note

additive_order() is a synonym for order()

EXAMPLES:

sage: E = EllipticCurve([0,0,1,-1,0])
sage: P = E([0,0]); P
(0 : 0 : 1)
sage: P.order()
+Infinity

sage: E = EllipticCurve([0,1])
sage: P = E([-1,0])
sage: P.order()
2
sage: P.additive_order()
2
archimedean_local_height(v=None, prec=None, weighted=False)#

Compute the local height of self at the archimedean place \(v\).

INPUT:

  • self – a point on an elliptic curve over a number field \(K\).

  • v – a real or complex embedding of K, or None (default). If \(v\) is a real or complex embedding, return the local height of self at \(v\). If \(v\) is None, return the total archimedean contribution to the global height.

  • prec – integer, or None (default). The precision of the computation. If None, the precision is deduced from v.

  • weighted – boolean. If False (default), the height is normalised to be invariant under extension of \(K\). If True, return this normalised height multiplied by the local degree if \(v\) is a single place, or by the degree of \(K\) if \(v\) is None.

OUTPUT:

A real number. The normalisation is twice that in Silverman’s paper [Sil1988]. Note that this local height depends on the model of the curve.

ALGORITHM:

See [Sil1988], Section 4.

EXAMPLES:

Examples 1, 2, and 3 from [Sil1988]:

sage: K.<a> = QuadraticField(-2)
sage: E = EllipticCurve(K, [0,-1,1,0,0]); E
Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 over Number Field in a with defining polynomial x^2 + 2 with a = 1.414213562373095?*I
sage: P = E.lift_x(2+a); P
(a + 2 : 2*a + 1 : 1)
sage: P.archimedean_local_height(K.places(prec=170)[0]) / 2
0.45754773287523276736211210741423654346576029814695

sage: K.<i> = NumberField(x^2+1)
sage: E = EllipticCurve(K, [0,0,4,6*i,0]); E
Elliptic Curve defined by y^2 + 4*y = x^3 + 6*i*x over Number Field in i with defining polynomial x^2 + 1
sage: P = E((0,0))
sage: P.archimedean_local_height(K.places()[0]) / 2
0.510184995162373

sage: Q = E.lift_x(-9/4); Q
(-9/4 : -27/8*i : 1)
sage: Q.archimedean_local_height(K.places()[0]) / 2
0.654445619529600

An example over the rational numbers:

sage: E = EllipticCurve([0, 0, 0, -36, 0])
sage: P = E([-3, 9])
sage: P.archimedean_local_height()
1.98723816350773

Local heights of torsion points can be non-zero (unlike the global height):

sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([0, 0, 0, K(1), 0])
sage: P = E(i, 0)
sage: P.archimedean_local_height()
0.346573590279973
elliptic_logarithm(embedding=None, precision=100, algorithm='pari')#

Return the elliptic logarithm of this elliptic curve point.

An embedding of the base field into \(\RR\) or \(\CC\) (with arbitrary precision) may be given; otherwise the first real embedding is used (with the specified precision) if any, else the first complex embedding.

INPUT:

  • embedding: an embedding of the base field into \(\RR\) or \(\CC\)

  • precision: a positive integer (default 100) setting the number of bits of precision for the computation

  • algorithm: either ‘pari’ (default for real embeddings) to use PARI’s pari:ellpointtoz, or ‘sage’ for a native implementation. Ignored for complex embeddings.

ALGORITHM:

See [Coh1993] for the case of real embeddings, and Cremona, J.E. and Thongjunthug, T. 2010 for the complex case.

AUTHORS:

  • Michael Mardaus (2008-07),

  • Tobias Nagel (2008-07) – original version from [Coh1993].

  • John Cremona (2008-07) – revision following eclib code.

  • John Cremona (2010-03) – implementation for complex embeddings.

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: E.discriminant() > 0
True
sage: P = E([-1,1])
sage: P.is_on_identity_component ()
False
sage: P.elliptic_logarithm (precision=96)
0.4793482501902193161295330101 + 0.985868850775824102211203849...*I
sage: Q=E([3,5])
sage: Q.is_on_identity_component()
True
sage: Q.elliptic_logarithm (precision=96)
1.931128271542559442488585220

An example with negative discriminant, and a torsion point:

sage: E = EllipticCurve('11a1')
sage: E.discriminant() < 0
True
sage: P = E([16,-61])
sage: P.elliptic_logarithm(precision=70)
0.25384186085591068434
sage: E.period_lattice().real_period(prec=70) / P.elliptic_logarithm(precision=70)
5.0000000000000000000

A larger example. The default algorithm uses PARI and makes sure the result has the requested precision:

sage: E = EllipticCurve([1, 0, 1, -85357462, 303528987048]) #18074g1
sage: P = E([4458713781401/835903744, -64466909836503771/24167649046528, 1])
sage: P.elliptic_logarithm()  # 100 bits
0.27656204014107061464076203097

The native algorithm ‘sage’ used to have trouble with precision in this example, but no longer:

sage: P.elliptic_logarithm(algorithm='sage')  # 100 bits
0.27656204014107061464076203097

This shows that the bug reported at trac ticket #4901 has been fixed:

sage: E = EllipticCurve("4390c2")
sage: P = E(683762969925/44944,-565388972095220019/9528128)
sage: P.elliptic_logarithm()
0.00025638725886520225353198932529
sage: P.elliptic_logarithm(precision=64)
0.000256387258865202254
sage: P.elliptic_logarithm(precision=65)
0.0002563872588652022535
sage: P.elliptic_logarithm(precision=128)
0.00025638725886520225353198932528666427412
sage: P.elliptic_logarithm(precision=129)
0.00025638725886520225353198932528666427412
sage: P.elliptic_logarithm(precision=256)
0.0002563872588652022535319893252866642741168388008346370015005142128009610936373
sage: P.elliptic_logarithm(precision=257)
0.00025638725886520225353198932528666427411683880083463700150051421280096109363730

Examples over number fields:

sage: K.<a> = NumberField(x^3-2)
sage: embs = K.embeddings(CC)
sage: E = EllipticCurve([0,1,0,a,a])
sage: Ls = [E.period_lattice(e) for e in embs]
sage: [L.real_flag for L in Ls]
[0, 0, -1]
sage: P = E(-1,0)  # order 2
sage: [L.elliptic_logarithm(P) for L in Ls]
[-1.73964256006716 - 1.07861534489191*I, -0.363756518406398 - 1.50699412135253*I, 1.90726488608927]

sage: E = EllipticCurve([-a^2 - a - 1, a^2 + a])
sage: Ls = [E.period_lattice(e) for e in embs]
sage: pts = [E(2*a^2 - a - 1 , -2*a^2 - 2*a + 6 ), E(-2/3*a^2 - 1/3 , -4/3*a - 2/3 ), E(5/4*a^2 - 1/2*a , -a^2 - 1/4*a + 9/4 ), E(2*a^2 + 3*a + 4 , -7*a^2 - 10*a - 12 )]
sage: [[L.elliptic_logarithm(P) for P in pts] for L in Ls]
[[0.250819591818930 - 0.411963479992219*I, -0.290994550611374 - 1.37239400324105*I, -0.693473752205595 - 2.45028458830342*I, -0.151659609775291 - 1.48985406505459*I], [1.33444787667954 - 1.50889756650544*I, 0.792633734249234 - 0.548467043256610*I, 0.390154532655013 + 0.529423541805758*I, 0.931968675085317 - 0.431006981443071*I], [1.14758249500109 + 0.853389664016075*I, 2.59823462472518 + 0.853389664016075*I, 1.75372176444709, 0.303069634723001]]

sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([0,0,0,9*i-10,21-i])
sage: emb = K.embeddings(CC)[1]
sage: L = E.period_lattice(emb)
sage: P = E(2-i,4+2*i)
sage: L.elliptic_logarithm(P,prec=100)
0.70448375537782208460499649302 - 0.79246725643650979858266018068*I
has_finite_order()#

Return True iff this point has finite order on the elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([0,0,1,-1,0])
sage: P = E([0,0]); P
(0 : 0 : 1)
sage: P.has_finite_order()
False

sage: E = EllipticCurve([0,1])
sage: P = E([-1,0])
sage: P.has_finite_order()
True
has_good_reduction(P=None)#

Returns True iff this point has good reduction modulo a prime.

INPUT:

  • P – a prime of the base_field of the point’s curve, or None (default)

OUTPUT:

(bool) If a prime \(P\) of the base field is specified, returns True iff the point has good reduction at \(P\); otherwise, return true if the point has god reduction at all primes in the support of the discriminant of this model.

EXAMPLES:

sage: E = EllipticCurve('990e1')
sage: P = E.gen(0); P
(15 : 51 : 1)
sage: [E.has_good_reduction(p) for p in [2,3,5,7]]
[False, False, False, True]
sage: [P.has_good_reduction(p) for p in [2,3,5,7]]
[True, False, True, True]
sage: [E.tamagawa_exponent(p) for p in [2,3,5,7]]
[2, 2, 1, 1]
sage: [(2*P).has_good_reduction(p) for p in [2,3,5,7]]
[True, True, True, True]
sage: P.has_good_reduction()
False
sage: (2*P).has_good_reduction()
True
sage: (3*P).has_good_reduction()
False

sage: K.<i> = NumberField(x^2+1)
sage: E = EllipticCurve(K,[0,1,0,-160,308])
sage: P = E(26,-120)
sage: E.discriminant().support()
[Fractional ideal (i + 1),
Fractional ideal (-i - 2),
Fractional ideal (2*i + 1),
Fractional ideal (3)]
sage: [E.tamagawa_exponent(p) for p in E.discriminant().support()]
[1, 4, 4, 4]
sage: P.has_good_reduction()
False
sage: (2*P).has_good_reduction()
False
sage: (4*P).has_good_reduction()
True
has_infinite_order()#

Return True iff this point has infinite order on the elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([0,0,1,-1,0])
sage: P = E([0,0]); P
(0 : 0 : 1)
sage: P.has_infinite_order()
True

sage: E = EllipticCurve([0,1])
sage: P = E([-1,0])
sage: P.has_infinite_order()
False
height(precision=None, normalised=True, algorithm='pari')#

Return the Néron-Tate canonical height of the point.

INPUT:

  • self – a point on an elliptic curve over a number field \(K\).

  • precision – positive integer, or None (default). The precision in bits of the result. If None, the default real precision is used.

  • normalised – boolean. If True (default), the height is normalised to be invariant under extension of \(K\). If False, return this normalised height multiplied by the degree of \(K\).

  • algorithm – string: either ‘pari’ (default) or ‘sage’. If ‘pari’ and the base field is \(\QQ\), use the PARI library function; otherwise use the Sage implementation.

OUTPUT:

The rational number 0, or a non-negative real number.

There are two normalisations used in the literature, one of which is double the other. We use the larger of the two, which is the one appropriate for the BSD conjecture. This is consistent with [Cre1997] and double that of [Sil2009].

See Wikipedia article Néron-Tate height.

Note

The correct height to use for the regulator in the BSD formula is the non-normalised height.

EXAMPLES:

sage: E = EllipticCurve('11a'); E
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sage: P = E([5,5]); P
(5 : 5 : 1)
sage: P.height()
0
sage: Q = 5*P
sage: Q.height()
0

sage: E = EllipticCurve('37a'); E
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: P = E([0,0])
sage: P.height()
0.0511114082399688
sage: P.order()
+Infinity
sage: E.regulator()
0.0511114082399688...

sage: def naive_height(P):
....:     return log(RR(max(abs(P[0].numerator()), abs(P[0].denominator()))))
sage: for n in [1..10]:
....:     print(naive_height(2^n*P)/4^n)
0.000000000000000
0.0433216987849966
0.0502949347635656
0.0511006335618645
0.0511007834799612
0.0511013666152466
0.0511034199907743
0.0511106492906471
0.0511114081541082
0.0511114081541180

sage: E = EllipticCurve('4602a1'); E
Elliptic Curve defined by y^2 + x*y  = x^3 + x^2 - 37746035*x - 89296920339 over Rational Field
sage: x = 77985922458974949246858229195945103471590
sage: y = 19575260230015313702261379022151675961965157108920263594545223
sage: d = 2254020761884782243
sage: E([ x / d^2,  y / d^3 ]).height()
86.7406561381275

sage: E = EllipticCurve([17, -60, -120, 0, 0]); E
Elliptic Curve defined by y^2 + 17*x*y - 120*y = x^3 - 60*x^2 over Rational Field
sage: E([30, -90]).height()
0

sage: E = EllipticCurve('389a1'); E
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
sage: [P,Q] = [E(-1,1),E(0,-1)]
sage: P.height(precision=100)
0.68666708330558658572355210295
sage: (3*Q).height(precision=100)/Q.height(precision=100)
9.0000000000000000000000000000
sage: _.parent()
Real Field with 100 bits of precision

Canonical heights over number fields are implemented as well:

sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^3-2)
sage: E = EllipticCurve([a, 4]); E
Elliptic Curve defined by y^2 = x^3 + a*x + 4 over Number Field in a with defining polynomial x^3 - 2
sage: P = E((0,2))
sage: P.height()
0.810463096585925
sage: P.height(precision=100)
0.81046309658592536863991810577
sage: P.height(precision=200)
0.81046309658592536863991810576865158896130286417155832378086
sage: (2*P).height() / P.height()
4.00000000000000
sage: (100*P).height() / P.height()
10000.0000000000

Setting normalised=False multiplies the height by the degree of \(K\):

sage: E = EllipticCurve('37a')
sage: P = E([0,0])
sage: P.height()
0.0511114082399688
sage: P.height(normalised=False)
0.0511114082399688
sage: K.<z> = CyclotomicField(5)
sage: EK = E.change_ring(K)
sage: PK = EK([0,0])
sage: PK.height()
0.0511114082399688
sage: PK.height(normalised=False)
0.204445632959875

Some consistency checks:

sage: E = EllipticCurve('5077a1')
sage: P = E([-2,3,1])
sage: P.height()
1.36857250535393

sage: EK = E.change_ring(QuadraticField(-3,'a'))
sage: PK = EK([-2,3,1])
sage: PK.height()
1.36857250535393

sage: K.<i> = NumberField(x^2+1)
sage: E = EllipticCurve(K, [0,0,4,6*i,0])
sage: Q = E.lift_x(-9/4); Q
(-9/4 : -27/8*i : 1)
sage: Q.height()
2.69518560017909
sage: (15*Q).height() / Q.height()
225.000000000000

sage: E = EllipticCurve('37a')
sage: P = E([0,-1])
sage: P.height()
0.0511114082399688
sage: K.<a> = QuadraticField(-7)
sage: ED = E.quadratic_twist(-7)
sage: Q = E.isomorphism_to(ED.change_ring(K))(P); Q
(0 : -7/2*a - 1/2 : 1)
sage: Q.height()
0.0511114082399688
sage: Q.height(precision=100)
0.051111408239968840235886099757

An example to show that the bug at trac ticket #5252 is fixed:

sage: E = EllipticCurve([1, -1, 1, -2063758701246626370773726978, 32838647793306133075103747085833809114881])
sage: P = E([-30987785091199, 258909576181697016447])
sage: P.height()
25.8603170675462
sage: P.height(precision=100)
25.860317067546190743868840741
sage: P.height(precision=250)
25.860317067546190743868840740735110323098872903844416215577171041783572513
sage: P.height(precision=500)
25.8603170675461907438688407407351103230988729038444162155771710417835725129551130570889813281792157278507639909972112856019190236125362914195452321720

sage: P.height(precision=100) == P.non_archimedean_local_height(prec=100)+P.archimedean_local_height(prec=100)
True

An example to show that the bug at trac ticket #8319 is fixed (correct height when the curve is not minimal):

sage: E = EllipticCurve([-5580472329446114952805505804593498080000,-157339733785368110382973689903536054787700497223306368000000])
sage: xP = 204885147732879546487576840131729064308289385547094673627174585676211859152978311600/23625501907057948132262217188983681204856907657753178415430361
sage: P = E.lift_x(xP)
sage: P.height()
157.432598516754
sage: Q = 2*P
sage: Q.height() # long time (4s)
629.730394067016
sage: Q.height()-4*P.height() # long time
0.000000000000000

An example to show that the bug at trac ticket #12509 is fixed (precision issues):

sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^2-x-1)
sage: v = [0, a + 1, 1, 28665*a - 46382, 2797026*a - 4525688]
sage: E = EllipticCurve(v)
sage: P = E([72*a - 509/5,  -682/25*a - 434/25])
sage: P.height()
1.38877711688727
sage: (2*P).height()/P.height()
4.00000000000000
sage: (2*P).height(precision=100)/P.height(precision=100)
4.0000000000000000000000000000
sage: (2*P).height(precision=1000)/P.height(precision=1000)
4.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

This shows that the bug reported at trac ticket #13951 has been fixed:

sage: E = EllipticCurve([0,17])
sage: P1 = E(2,5)
sage: P1.height()
1.06248137652528
sage: F = E.change_ring(QuadraticField(-3,'a'))
sage: P2 = F([2,5])
sage: P2.height()
1.06248137652528
is_on_identity_component(embedding=None)#

Returns True iff this point is on the identity component of its curve with respect to a given (real or complex) embedding.

INPUT:

  • self – a point on a curve over any ordered field (e.g. \(\QQ\))

  • embedding – an embedding from the base_field of the point’s curve into \(\RR\) or \(\CC\); if None (the default) it uses the first embedding of the base_field into \(\RR\) if any, else the first embedding into \(\CC\).

OUTPUT:

(bool) – True iff the point is on the identity component of the curve. (If the point is zero then the result is True.)

EXAMPLES:

For \(K=\QQ\) there is no need to specify an embedding:

sage: E=EllipticCurve('5077a1')
sage: [E.lift_x(x).is_on_identity_component() for x in srange(-3,5)]
[False, False, False, False, False, True, True, True]

An example over a field with two real embeddings:

sage: L.<a> = QuadraticField(2)
sage: E=EllipticCurve(L,[0,1,0,a,a])
sage: P=E(-1,0)
sage: [P.is_on_identity_component(e) for e in L.embeddings(RR)]
[False, True]

We can check this as follows:

sage: [e(E.discriminant())>0 for e in L.embeddings(RR)]
[True, False]
sage: e = L.embeddings(RR)[0]
sage: E1 = EllipticCurve(RR,[e(ai) for ai in E.ainvs()])
sage: e1,e2,e3 = E1.two_division_polynomial().roots(RR,multiplicities=False)
sage: e1 < e2 < e3 and e(P[0]) < e3
True
non_archimedean_local_height(v=None, prec=None, weighted=False, is_minimal=None)#

Compute the local height of self at the non-archimedean place \(v\).

INPUT:

  • self – a point on an elliptic curve over a number field \(K\).

  • v – a non-archimedean place of \(K\), or None (default). If \(v\) is a non-archimedean place, return the local height of self at \(v\). If \(v\) is None, return the total non-archimedean contribution to the global height.

  • prec – integer, or None (default). The precision of the computation. If None, the height is returned symbolically.

  • weighted – boolean. If False (default), the height is normalised to be invariant under extension of \(K\). If True, return this normalised height multiplied by the local degree if \(v\) is a single place, or by the degree of \(K\) if \(v\) is None.

OUTPUT:

A real number. The normalisation is twice that in Silverman’s paper [Sil1988]. Note that this local height depends on the model of the curve.

ALGORITHM:

See [Sil1988], Section 5.

EXAMPLES:

Examples 2 and 3 from [Sil1988]:

sage: K.<i> = NumberField(x^2+1)
sage: E = EllipticCurve(K, [0,0,4,6*i,0]); E
Elliptic Curve defined by y^2 + 4*y = x^3 + 6*i*x over Number Field in i with defining polynomial x^2 + 1
sage: P = E((0,0))
sage: P.non_archimedean_local_height(K.ideal(i+1))
-1/2*log(2)
sage: P.non_archimedean_local_height(K.ideal(3))
0
sage: P.non_archimedean_local_height(K.ideal(1-2*i))
0

sage: Q = E.lift_x(-9/4); Q
(-9/4 : -27/8*i : 1)
sage: Q.non_archimedean_local_height(K.ideal(1+i))
2*log(2)
sage: Q.non_archimedean_local_height(K.ideal(3))
0
sage: Q.non_archimedean_local_height(K.ideal(1-2*i))
0
sage: Q.non_archimedean_local_height()
2*log(2)

An example over the rational numbers:

sage: E = EllipticCurve([0, 0, 0, -36, 0])
sage: P = E([-3, 9])
sage: P.non_archimedean_local_height()
-log(3)

Local heights of torsion points can be non-zero (unlike the global height):

sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([0, 0, 0, K(1), 0])
sage: P = E(i, 0)
sage: P.non_archimedean_local_height()
-1/2*log(2)
order()#

Return the order of this point on the elliptic curve.

If the point has infinite order, returns +Infinity. For curves defined over \(\QQ\), we call PARI; over other number fields we implement the function here.

Note

additive_order() is a synonym for order()

EXAMPLES:

sage: E = EllipticCurve([0,0,1,-1,0])
sage: P = E([0,0]); P
(0 : 0 : 1)
sage: P.order()
+Infinity

sage: E = EllipticCurve([0,1])
sage: P = E([-1,0])
sage: P.order()
2
sage: P.additive_order()
2
padic_elliptic_logarithm(p, absprec=20)#

Computes the \(p\)-adic elliptic logarithm of this point.

INPUT:

p - integer: a prime absprec - integer (default: 20): the initial \(p\)-adic absolute precision of the computation

OUTPUT:

The \(p\)-adic elliptic logarithm of self, with precision absprec.

AUTHORS:

  • Tobias Nagel

  • Michael Mardaus

  • John Cremona

ALGORITHM:

For points in the formal group (i.e. not integral at \(p\)) we take the log() function from the formal groups module and evaluate it at \(-x/y\). Otherwise we first multiply the point to get into the formal group, and divide the result afterwards.

Todo

See comments at trac ticket #4805. Currently the absolute precision of the result may be less than the given value of absprec, and error-handling is imperfect.

EXAMPLES:

sage: E = EllipticCurve([0,1,1,-2,0])
sage: E(0).padic_elliptic_logarithm(3)
0
sage: P = E(0,0)
sage: P.padic_elliptic_logarithm(3)
2 + 2*3 + 3^3 + 2*3^7 + 3^8 + 3^9 + 3^11 + 3^15 + 2*3^17 + 3^18 + O(3^19)
sage: P.padic_elliptic_logarithm(3).lift()
660257522
sage: P = E(-11/9,28/27)
sage: [(2*P).padic_elliptic_logarithm(p)/P.padic_elliptic_logarithm(p) for p in prime_range(20)]  # long time (3s)
[2 + O(2^19), 2 + O(3^20), 2 + O(5^19), 2 + O(7^19), 2 + O(11^19), 2 + O(13^19), 2 + O(17^19), 2 + O(19^19)]
sage: [(3*P).padic_elliptic_logarithm(p)/P.padic_elliptic_logarithm(p) for p in prime_range(12)]  # long time (2s)
[1 + 2 + O(2^19), 3 + 3^20 + O(3^21), 3 + O(5^19), 3 + O(7^19), 3 + O(11^19)]
sage: [(5*P).padic_elliptic_logarithm(p)/P.padic_elliptic_logarithm(p) for p in prime_range(12)]  # long time (2s)
[1 + 2^2 + O(2^19), 2 + 3 + O(3^20), 5 + O(5^19), 5 + O(7^19), 5 + O(11^19)]

An example which arose during reviewing trac ticket #4741:

sage: E = EllipticCurve('794a1')
sage: P = E(-1,2)
sage: P.padic_elliptic_logarithm(2) # default precision=20
2^4 + 2^5 + 2^6 + 2^8 + 2^9 + 2^13 + 2^14 + 2^15 + O(2^16)
sage: P.padic_elliptic_logarithm(2, absprec=30)
2^4 + 2^5 + 2^6 + 2^8 + 2^9 + 2^13 + 2^14 + 2^15 + 2^22 + 2^23 + 2^24 + O(2^26)
sage: P.padic_elliptic_logarithm(2, absprec=40)
2^4 + 2^5 + 2^6 + 2^8 + 2^9 + 2^13 + 2^14 + 2^15 + 2^22 + 2^23 + 2^24 + 2^28 + 2^29 + 2^31 + 2^34 + O(2^35)
reduction(p)#

This finds the reduction of a point \(P\) on the elliptic curve modulo the prime \(p\).

INPUT:

  • self – A point on an elliptic curve.

  • p – a prime number

OUTPUT:

The point reduced to be a point on the elliptic curve modulo \(p\).

EXAMPLES:

sage: E = EllipticCurve([1,2,3,4,0])
sage: P = E(0,0)
sage: P.reduction(5)
(0 : 0 : 1)
sage: Q = E(98,931)
sage: Q.reduction(5)
(3 : 1 : 1)
sage: Q.reduction(5).curve() == E.reduction(5)
True

sage: F.<a> = NumberField(x^2+5)
sage: E = EllipticCurve(F,[1,2,3,4,0])
sage: Q = E(98,931)
sage: Q.reduction(a)
(3 : 1 : 1)
sage: Q.reduction(11)
(10 : 7 : 1)

sage: F.<a> = NumberField(x^3+x^2+1)
sage: E = EllipticCurve(F,[a,2])
sage: P = E(a,1)
sage: P.reduction(F.ideal(5))
(abar : 1 : 1)
sage: P.reduction(F.ideal(a^2-4*a-2))
(abar : 1 : 1)