Heegner points on elliptic curves over the rational numbers

AUTHORS:

  • William Stein (August 2009)– most of the initial version
  • Robert Bradshaw (July 2009) – an early version of some specific code

EXAMPLES:

sage: E = EllipticCurve('433a')
sage: P = E.heegner_point(-8,3)
sage: z = P.point_exact(201); z
(-4/3 : 1/27*a - 4/27 : 1)
sage: parent(z)
Abelian group of points on Elliptic Curve defined by y^2 + x*y = x^3 + 1 over Number Field in a with defining polynomial x^2 - 44*x + 1159
sage: parent(z[0]).discriminant()
-3
sage: E.quadratic_twist(-3).rank()
1
sage: K.<a> = QuadraticField(-8)
sage: K.factor(3)
(Fractional ideal (1/2*a + 1)) * (Fractional ideal (1/2*a - 1))

Next try an inert prime:

sage: K.factor(5)
Fractional ideal (5)
sage: P = E.heegner_point(-8,5)
sage: z = P.point_exact(300)
sage: z[0].charpoly().factor()
(x^6 + x^5 - 1/4*x^4 + 19/10*x^3 + 31/20*x^2 - 7/10*x + 49/100)^2
sage: z[1].charpoly().factor()
x^12 - x^11 + 6/5*x^10 - 33/40*x^9 - 89/320*x^8 + 3287/800*x^7 - 5273/1600*x^6 + 993/4000*x^5 + 823/320*x^4 - 2424/625*x^3 + 12059/12500*x^2 + 3329/25000*x + 123251/250000
sage: f = P.x_poly_exact(300); f
x^6 + x^5 - 1/4*x^4 + 19/10*x^3 + 31/20*x^2 - 7/10*x + 49/100
sage: f.discriminant().factor()
-1 * 2^-9 * 5^-9 * 7^2 * 281^2 * 1021^2

We find some Mordell-Weil generators in the rank 1 case using Heegner points:

sage: E = EllipticCurve('43a'); P = E.heegner_point(-7)
sage: P.x_poly_exact()
x
sage: P.point_exact()
(0 : 0 : 1)

sage: E = EllipticCurve('997a')
sage: E.rank()
1
sage: E.heegner_discriminants_list(10)
[-19, -23, -31, -35, -39, -40, -52, -55, -56, -59]
sage: P = E.heegner_point(-19)
sage: P.x_poly_exact()
x - 141/49
sage: P.point_exact()
(141/49 : -162/343 : 1)

Here we find that the Heegner point generates a subgroup of index 3:

sage: E = EllipticCurve('92b1')
sage: E.heegner_discriminants_list(1)
[-7]
sage: P = E.heegner_point(-7); z = P.point_exact(); z
(0 : 1 : 1)
sage: E.regulator()
0.0498083972980648
sage: z.height()
0.448275575682583
sage: P = E(1,1); P # a generator
(1 : 1 : 1)
sage: -3*P
(0 : 1 : 1)
sage: E.tamagawa_product()
3

The above is consistent with the following analytic computation:

sage: E.heegner_index(-7)
3.0000?
class sage.schemes.elliptic_curves.heegner.GaloisAutomorphism(parent)

Bases: sage.structure.sage_object.SageObject

An abstract automorphism of a ring class field.

Todo

make GaloisAutomorphism derive from GroupElement, so that one gets powers for free, etc.

domain()

Return the domain of this automorphism.

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: s = E.heegner_point(-7,5).ring_class_field().galois_group().complex_conjugation()
sage: s.domain()
Ring class field extension of QQ[sqrt(-7)] of conductor 5
parent()

Return the parent of this automorphism, which is a Galois group of a ring class field.

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: s = E.heegner_point(-7,5).ring_class_field().galois_group().complex_conjugation()
sage: s.parent()
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
class sage.schemes.elliptic_curves.heegner.GaloisAutomorphismComplexConjugation(parent)

Bases: sage.schemes.elliptic_curves.heegner.GaloisAutomorphism

The complex conjugation automorphism of a ring class field.

EXAMPLES:

sage: conj = heegner_point(37,-7,5).ring_class_field().galois_group().complex_conjugation()
sage: conj
Complex conjugation automorphism of Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: conj.domain()
Ring class field extension of QQ[sqrt(-7)] of conductor 5

TESTS:

sage: type(conj)
<class 'sage.schemes.elliptic_curves.heegner.GaloisAutomorphismComplexConjugation'>
sage: loads(dumps(conj)) == conj
True
order()

EXAMPLES:

sage: conj = heegner_point(37,-7,5).ring_class_field().galois_group().complex_conjugation()
sage: conj.order()
2
class sage.schemes.elliptic_curves.heegner.GaloisAutomorphismQuadraticForm(parent, quadratic_form, alpha=None)

Bases: sage.schemes.elliptic_curves.heegner.GaloisAutomorphism

An automorphism of a ring class field defined by a quadratic form.

EXAMPLES:

sage: H = heegner_points(389,-20,3)
sage: sigma = H.ring_class_field().galois_group(H.quadratic_field())[0]; sigma
Class field automorphism defined by x^2 + 45*y^2
sage: type(sigma)
<class 'sage.schemes.elliptic_curves.heegner.GaloisAutomorphismQuadraticForm'>
sage: loads(dumps(sigma)) == sigma
True
alpha()

Optional data that specified element corresponding element of \((\mathcal{O}_K / c\mathcal{O}_K)^* / (\ZZ/c\ZZ)^*\), via class field theory.

This is a generator of the ideal corresponding to this automorphism.

EXAMPLES:

sage: K3 = heegner_points(389,-52,3).ring_class_field()
sage: K1 = heegner_points(389,-52,1).ring_class_field()
sage: G = K3.galois_group(K1)
sage: orb = sorted([g.alpha() for g in G]); orb # random (the sign depends on the database being installed or not)
[1, 1/2*sqrt_minus_52 + 1, -1/2*sqrt_minus_52, 1/2*sqrt_minus_52 - 1]
sage: sorted([x^2 for x in orb]) # this is just for testing
[-13, -sqrt_minus_52 - 12, sqrt_minus_52 - 12, 1]

sage: K5 = heegner_points(389,-52,5).ring_class_field()
sage: K1 = heegner_points(389,-52,1).ring_class_field()
sage: G = K5.galois_group(K1)
sage: orb = sorted([g.alpha() for g in G]); orb # random (the sign depends on the database being installed or not)
[1, -1/2*sqrt_minus_52, 1/2*sqrt_minus_52 + 1, 1/2*sqrt_minus_52 - 1, 1/2*sqrt_minus_52 - 2, -1/2*sqrt_minus_52 - 2]
sage: sorted([x^2 for x in orb]) # just for testing
[-13, -sqrt_minus_52 - 12, sqrt_minus_52 - 12, -2*sqrt_minus_52 - 9, 2*sqrt_minus_52 - 9, 1]
ideal()

Return ideal of ring of integers of quadratic imaginary field corresponding to this quadratic form. This is the ideal

\(I = \left(A, \frac{-B+ c\sqrt{D}}{2}\right) \mathcal{O}_K\).

EXAMPLES:

sage: E = EllipticCurve('389a'); F= E.heegner_point(-20,3).ring_class_field()
sage: G = F.galois_group(F.quadratic_field())
sage: G[1].ideal()
Fractional ideal (2, 1/2*sqrt_minus_20 + 1)
sage: [s.ideal().gens() for s in G]
[(1, 3/2*sqrt_minus_20), (2, 3/2*sqrt_minus_20 - 1), (5, 3/2*sqrt_minus_20), (7, 3/2*sqrt_minus_20 - 2)]
order()

Return the multiplicative order of this Galois group automorphism.

EXAMPLES:

sage: K3 = heegner_points(389,-52,3).ring_class_field()
sage: K1 = heegner_points(389,-52,1).ring_class_field()
sage: G = K3.galois_group(K1)
sage: sorted([g.order() for g in G])
[1, 2, 4, 4]
sage: K5 = heegner_points(389,-52,5).ring_class_field()
sage: K1 = heegner_points(389,-52,1).ring_class_field()
sage: G = K5.galois_group(K1)
sage: sorted([g.order() for g in G])
[1, 2, 3, 3, 6, 6]
p1_element()

Return element of the projective line corresponding to this automorphism.

This only makes sense if this automorphism is in the Galois group \(\textrm{Gal}(K_c/K_1)\).

EXAMPLES:

sage: K3 = heegner_points(389,-52,3).ring_class_field()
sage: K1 = heegner_points(389,-52,1).ring_class_field()
sage: G = K3.galois_group(K1)
sage: sorted([g.p1_element() for g in G])
[(0, 1), (1, 0), (1, 1), (1, 2)]

sage: K5 = heegner_points(389,-52,5).ring_class_field()
sage: K1 = heegner_points(389,-52,1).ring_class_field()
sage: G = K5.galois_group(K1)
sage: sorted([g.p1_element() for g in G])
[(0, 1), (1, 0), (1, 1), (1, 2), (1, 3), (1, 4)]
quadratic_form()

Return reduced quadratic form corresponding to this Galois automorphism.

EXAMPLES:

sage: H = heegner_points(389,-20,3); s = H.ring_class_field().galois_group(H.quadratic_field())[0]
sage: s.quadratic_form()
x^2 + 45*y^2
class sage.schemes.elliptic_curves.heegner.GaloisGroup(field, base=Rational Field)

Bases: sage.structure.sage_object.SageObject

A Galois group of a ring class field.

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: G = E.heegner_point(-7,5).ring_class_field().galois_group(); G
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: G.field()
Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: G.cardinality()
12
sage: G.complex_conjugation()
Complex conjugation automorphism of Ring class field extension of QQ[sqrt(-7)] of conductor 5

TESTS:

sage: G = heegner_point(37,-7).ring_class_field().galois_group()
sage: loads(dumps(G)) == G
True
sage: type(G)
<class 'sage.schemes.elliptic_curves.heegner.GaloisGroup'>
base_field()

Return the base field, which the field fixed by all the automorphisms in this Galois group.

EXAMPLES:

sage: x = heegner_point(37,-7,5)
sage: Kc = x.ring_class_field(); Kc
Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: K = x.quadratic_field()
sage: G = Kc.galois_group(); G
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: G.base_field()
Rational Field
sage: G.cardinality()
12
sage: Kc.absolute_degree()
12
sage: G = Kc.galois_group(K); G
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5 over Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
sage: G.cardinality()
6
sage: G.base_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
sage: G = Kc.galois_group(Kc); G
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5 over Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: G.cardinality()
1
sage: G.base_field()
Ring class field extension of QQ[sqrt(-7)] of conductor 5
cardinality()

Return the cardinality of this Galois group.

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: G = E.heegner_point(-7,5).ring_class_field().galois_group(); G
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: G.cardinality()
12
sage: G = E.heegner_point(-7).ring_class_field().galois_group()
sage: G.cardinality()
2
sage: G = E.heegner_point(-7,55).ring_class_field().galois_group()
sage: G.cardinality()
120
complex_conjugation()

Return the automorphism of self determined by complex conjugation. The base field must be the rational numbers.

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: G = E.heegner_point(-7,5).ring_class_field().galois_group()
sage: G.complex_conjugation()
Complex conjugation automorphism of Ring class field extension of QQ[sqrt(-7)] of conductor 5
field()

Return the ring class field that this Galois group acts on.

EXAMPLES:

sage: G = heegner_point(389,-7,5).ring_class_field().galois_group()
sage: G.field()
Ring class field extension of QQ[sqrt(-7)] of conductor 5
is_kolyvagin()

Return True if conductor \(c\) is prime to the discriminant of the quadratic field, \(c\) is squarefree and each prime dividing \(c\) is inert.

EXAMPLES:

sage: K5 = heegner_points(389,-52,5).ring_class_field()
sage: K1 = heegner_points(389,-52,1).ring_class_field()
sage: K5.galois_group(K1).is_kolyvagin()
True
sage: K7 = heegner_points(389,-52,7).ring_class_field()
sage: K7.galois_group(K1).is_kolyvagin()
False
sage: K25 = heegner_points(389,-52,25).ring_class_field()
sage: K25.galois_group(K1).is_kolyvagin()
False
kolyvagin_generators()

Assuming this Galois group \(G\) is of the form \(G=\textrm{Gal}(K_c/K_1)\), with \(c=p_1\dots p_n\) satisfying the Kolyvagin hypothesis, this function returns noncanonical choices of lifts of generators for each of the cyclic factors of \(G\) corresponding to the primes dividing \(c\). Thus the \(i\)-th returned valued is an element of \(G\) that maps to the identity element of \(\textrm{Gal}(K_p/K_1)\) for all \(p \neq p_i\) and to a choice of generator of \(\textrm{Gal}(K_{p_i}/K_1)\).

OUTPUT:

  • list of elements of self

EXAMPLES:

sage: K3 = heegner_points(389,-52,3).ring_class_field()
sage: K1 = heegner_points(389,-52,1).ring_class_field()
sage: G = K3.galois_group(K1)
sage: G.kolyvagin_generators()
(Class field automorphism defined by 9*x^2 - 6*x*y + 14*y^2,)

sage: K5 = heegner_points(389,-52,5).ring_class_field()
sage: K1 = heegner_points(389,-52,1).ring_class_field()
sage: G = K5.galois_group(K1)
sage: G.kolyvagin_generators()
(Class field automorphism defined by 17*x^2 - 14*x*y + 22*y^2,)
lift_of_hilbert_class_field_galois_group()

Assuming this Galois group \(G\) is of the form \(G=\textrm{Gal}(K_c/K)\), this function returns noncanonical choices of lifts of the elements of the quotient group \(\textrm{Gal}(K_1/K)\).

OUTPUT:

  • tuple of elements of self

EXAMPLES:

sage: K5 = heegner_points(389,-52,5).ring_class_field()
sage: G = K5.galois_group(K5.quadratic_field())
sage: G.lift_of_hilbert_class_field_galois_group()
(Class field automorphism defined by x^2 + 325*y^2, Class field automorphism defined by 2*x^2 + 2*x*y + 163*y^2)
sage: G.cardinality()
12
sage: K5.quadratic_field().class_number()
2
class sage.schemes.elliptic_curves.heegner.HeegnerPoint(N, D, c)

Bases: sage.structure.sage_object.SageObject

A Heegner point of level \(N\), discriminant \(D\) and conductor \(c\) is any point on a modular curve or elliptic curve that is concocted in some way from a quadratic imaginary \(\tau\) in the upper half plane with \(\Delta(\tau) = D c = \Delta(N \tau)\).

EXAMPLES:

sage: x = sage.schemes.elliptic_curves.heegner.HeegnerPoint(389,-7,13); x
Heegner point of level 389, discriminant -7, and conductor 13
sage: type(x)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPoint'>
sage: loads(dumps(x)) == x
True
conductor()

Return the conductor of this Heegner point.

EXAMPLES:

sage: heegner_point(389,-7,5).conductor()
5
sage: E = EllipticCurve('37a1'); P = E.kolyvagin_point(-67,7); P
Kolyvagin point of discriminant -67 and conductor 7 on elliptic curve of conductor 37
sage: P.conductor()
7
sage: E = EllipticCurve('389a'); P = E.heegner_point(-7, 5); P.conductor()
5
discriminant()

Return the discriminant of the quadratic imaginary field associated to this Heegner point.

EXAMPLES:

sage: heegner_point(389,-7,5).discriminant()
-7
sage: E = EllipticCurve('37a1'); P = E.kolyvagin_point(-67,7); P
Kolyvagin point of discriminant -67 and conductor 7 on elliptic curve of conductor 37
sage: P.discriminant()
-67
sage: E = EllipticCurve('389a'); P = E.heegner_point(-7, 5); P.discriminant()
-7
level()

Return the level of this Heegner point, which is the level of the modular curve \(X_0(N)\) on which this is a Heegner point.

EXAMPLES:

sage: heegner_point(389,-7,5).level()
389
quadratic_field()

Return the quadratic number field of discriminant \(D\).

EXAMPLES:

sage: x = heegner_point(37,-7,5)
sage: x.quadratic_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7


sage: E = EllipticCurve('37a'); P = E.heegner_point(-40)
sage: P.quadratic_field()
Number Field in sqrt_minus_40 with defining polynomial x^2 + 40
sage: P.quadratic_field() is P.quadratic_field()
True
sage: type(P.quadratic_field())
<class 'sage.rings.number_field.number_field.NumberField_quadratic_with_category'>
quadratic_order()

Return the order in the quadratic imaginary field of conductor \(c\), where \(c\) is the conductor of this Heegner point.

EXAMPLES:

sage: heegner_point(389,-7,5).quadratic_order()
Order in Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
sage: heegner_point(389,-7,5).quadratic_order().basis()
[1, 5*sqrt_minus_7]

sage: E = EllipticCurve('37a'); P = E.heegner_point(-40,11)
sage: P.quadratic_order()
Order in Number Field in sqrt_minus_40 with defining polynomial x^2 + 40
sage: P.quadratic_order().basis()
[1, 11*sqrt_minus_40]
ring_class_field()

Return the ring class field associated to this Heegner point. This is an extension \(K_c\) over \(K\), where \(K\) is the quadratic imaginary field and \(c\) is the conductor associated to this Heegner point. This Heegner point is defined over \(K_c\) and the Galois group \(Gal(K_c/K)\) acts transitively on the Galois conjugates of this Heegner point.

EXAMPLES:

sage: E = EllipticCurve('389a'); K.<a> = QuadraticField(-5)
sage: len(K.factor(5))
1
sage: len(K.factor(23))
2
sage: E.heegner_point(-7, 5).ring_class_field().degree_over_K()
6
sage: E.heegner_point(-7, 23).ring_class_field().degree_over_K()
22
sage: E.heegner_point(-7, 5*23).ring_class_field().degree_over_K()
132
sage: E.heegner_point(-7, 5^2).ring_class_field().degree_over_K()
30
sage: E.heegner_point(-7, 7).ring_class_field().degree_over_K()
7
class sage.schemes.elliptic_curves.heegner.HeegnerPointOnEllipticCurve(E, x, check=True)

Bases: sage.schemes.elliptic_curves.heegner.HeegnerPoint

A Heegner point on a curve associated to an order in a quadratic imaginary field.

EXAMPLES:

sage: E = EllipticCurve('37a'); P = E.heegner_point(-7,5); P
Heegner point of discriminant -7 and conductor 5 on elliptic curve of conductor 37
sage: type(P)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPointOnEllipticCurve'>
conjugates_over_K()

Return the \(Gal(K_c/K)\) conjugates of this Heegner point.

EXAMPLES:

sage: E = EllipticCurve('77a')
sage: y = E.heegner_point(-52,5); y
Heegner point of discriminant -52 and conductor 5 on elliptic curve of conductor 77
sage: print [z.quadratic_form() for z in y.conjugates_over_K()]
[77*x^2 + 52*x*y + 13*y^2, 154*x^2 + 206*x*y + 71*y^2, 539*x^2 + 822*x*y + 314*y^2, 847*x^2 + 1284*x*y + 487*y^2, 1001*x^2 + 52*x*y + y^2, 1078*x^2 + 822*x*y + 157*y^2, 1309*x^2 + 360*x*y + 25*y^2, 1309*x^2 + 2054*x*y + 806*y^2, 1463*x^2 + 976*x*y + 163*y^2, 2233*x^2 + 2824*x*y + 893*y^2, 2387*x^2 + 2054*x*y + 442*y^2, 3619*x^2 + 3286*x*y + 746*y^2]
sage: y.quadratic_form()
77*x^2 + 52*x*y + 13*y^2
curve()

Return the elliptic curve on which this is a Heegner point.

EXAMPLES:

sage: E = EllipticCurve('389a'); P = E.heegner_point(-7, 5)
sage: P.curve()
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
sage: P.curve() is E
True
heegner_point_on_X0N()

Return Heegner point on \(X_0(N)\) that maps to this Heegner point on \(E\).

EXAMPLES:

sage: E = EllipticCurve('37a'); P = E.heegner_point(-7,5); P
Heegner point of discriminant -7 and conductor 5 on elliptic curve of conductor 37
sage: P.heegner_point_on_X0N()
Heegner point 5/74*sqrt(-7) - 11/74 of discriminant -7 and conductor 5 on X_0(37)
kolyvagin_cohomology_class(n=None)

Return the Kolyvagin class associated to this Heegner point.

INPUT:

  • \(n\) – positive integer that divides the gcd of \(a_p\) and \(p+1\) for all \(p\) dividing the conductor. If \(n\) is None, choose the largest valid \(n\).

EXAMPLES:

sage: y = EllipticCurve('389a').heegner_point(-7,5)
sage: y.kolyvagin_cohomology_class(3)
Kolyvagin cohomology class c(5) in H^1(K,E[3])
kolyvagin_point()

Return the Kolyvagin point corresponding to this Heegner point. This is the point obtained by applying the Kolyvagin operator \(J_c I_c\) in the group ring of the Galois group to this Heegner point. It is a point that defines an element of \(H^1(K, E[n])\), under certain hypotheses on \(n\).

EXAMPLES:

sage: E = EllipticCurve('37a1'); y = E.heegner_point(-7); y
Heegner point of discriminant -7 on elliptic curve of conductor 37
sage: P = y.kolyvagin_point(); P
Kolyvagin point of discriminant -7 on elliptic curve of conductor 37
sage: PP = P.numerical_approx() # approximately (0 : 0 : 1)
sage: all([c.abs() < 1e-15 for c in PP.xy()])
True
map_to_complex_numbers(prec=53)

Return the point in the subfield \(M\) of the complex numbers (well defined only modulo the period lattice) corresponding to this Heegner point.

EXAMPLES:

We compute a nonzero Heegner point over a ring class field on a curve of rank 2:

sage: E = EllipticCurve('389a'); y = E.heegner_point(-7,5)
sage: y.map_to_complex_numbers()
1.49979679635196 + 0.369156204821526*I
sage: y.map_to_complex_numbers(100)
1.4997967963519640592142411892 + 0.36915620482152626830089145962*I
sage: y.map_to_complex_numbers(10)
1.5 + 0.37*I

Here we see that the Heegner point is 0 since it lies in the lattice:

sage: E = EllipticCurve('389a'); y = E.heegner_point(-7)
sage: y.map_to_complex_numbers(10)
0.0034 - 3.9*I
sage: y.map_to_complex_numbers()
4.71844785465692e-15 - 3.94347540310330*I
sage: E.period_lattice().basis()
(2.49021256085505, 1.97173770155165*I)
sage: 2*E.period_lattice().basis()[1]
3.94347540310330*I

You can also directly coerce to the complex field:

sage: E = EllipticCurve('389a'); y = E.heegner_point(-7)
sage: z = ComplexField(100)(y); z # real part approx. 0
-... - 3.9434754031032964088448153963*I
sage: E.period_lattice().elliptic_exponential(z)
(0.00000000000000000000000000000 : 1.0000000000000000000000000000 : 0.00000000000000000000000000000)
numerical_approx(prec=53, algorithm=None)

Return a numerical approximation to this Heegner point computed using a working precision of prec bits.

Warning

The answer is not provably correct to prec bits! A priori, due to rounding and other errors, it is possible that not a single digit is correct.

INPUT:

  • prec – (default: None) the working precision

EXAMPLES:

sage: E = EllipticCurve('37a'); P = E.heegner_point(-7); P
Heegner point of discriminant -7 on elliptic curve of conductor 37
sage: all([c.abs()< 1e-15 for c in P.numerical_approx().xy()])
True
sage: P.numerical_approx(10)  # expect random digits
(0.0030 - 0.0028*I : -0.0030 + 0.0028*I : 1.0)
sage: P.numerical_approx(100)[0]  # expect random digits
8.4...e-31 + 6.0...e-31*I
sage: E = EllipticCurve('37a'); P = E.heegner_point(-40); P
Heegner point of discriminant -40 on elliptic curve of conductor 37
sage: P.numerical_approx()
(-6.6...e-16 + 1.41421356237310*I : 1.00000000000000 - 1.41421356237309*I : 1.00000000000000)

A rank 2 curve, where all Heegner points of conductor 1 are 0:

sage: E = EllipticCurve('389a'); E.rank()
2
sage: P = E.heegner_point(-7); P
Heegner point of discriminant -7 on elliptic curve of conductor 389
sage: P.numerical_approx()
(0.000000000000000 : 1.00000000000000 : 0.000000000000000)

However, Heegner points of bigger conductor are often nonzero:

sage: E = EllipticCurve('389a'); P = E.heegner_point(-7, 5); P
Heegner point of discriminant -7 and conductor 5 on elliptic curve of conductor 389
sage: numerical_approx(P)
(0.675507556926806 + 0.344749649302635*I : -0.377142931401887 + 0.843366227137146*I : 1.00000000000000)
sage: P.numerical_approx()
(0.6755075569268... + 0.3447496493026...*I : -0.3771429314018... + 0.8433662271371...*I : 1.00000000000000)
sage: E.heegner_point(-7, 11).numerical_approx()
(0.1795583794118... + 0.02035501750912...*I : -0.5573941377055... + 0.2738940831635...*I : 1.00000000000000)
sage: E.heegner_point(-7, 13).numerical_approx()
(1.034302915374... - 3.302744319777...*I : 1.323937875767... + 6.908264226850...*I : 1.00000000000000)

We find (probably) the definining polynomial of the \(x\)-coordinate of \(P\), which defines a class field. The shape of the discriminant below is strong confirmation – but not proof – that this polynomial is correct:

sage: f = P.numerical_approx(70)[0].algdep(6); f
1225*x^6 + 1750*x^5 - 21675*x^4 - 380*x^3 + 110180*x^2 - 129720*x + 48771
sage: f.discriminant().factor()
2^6 * 3^2 * 5^11 * 7^4 * 13^2 * 19^6 * 199^2 * 719^2 * 26161^2
point_exact(prec=53, algorithm='lll', var='a', optimize=False)

Return exact point on the elliptic curve over a number field defined by computing this Heegner point to the given number of bits of precision. A ValueError is raised if the precision is clearly insignificant to define a point on the curve.

Warning

It is in theory possible for this function to not raise a ValueError, find a point on the curve, but via some very unlikely coincidence that point is not actually this Heegner point.

Warning

Currently we make an arbitrary choice of \(y\)-coordinate for the lift of the \(x\)-coordinate.

INPUT:

  • prec – integer (default: 53)
  • algorithm – see the description of the algorithm parameter for the x_poly_exact method.
  • var – string (default: ‘a’)
  • optimize – book (default; False) if True, try to optimize defining polynomial for the number field that the point is defined over. Off by default, since this can be very expensive.

EXAMPLES:

sage: E = EllipticCurve('389a'); P = E.heegner_point(-7, 5); P
Heegner point of discriminant -7 and conductor 5 on elliptic curve of conductor 389
sage: z = P.point_exact(100, optimize=True)
sage: z[1].charpoly()
x^12 + 6*x^11 + 90089/1715*x^10 + 71224/343*x^9 + 52563964/588245*x^8 - 483814934/588245*x^7 - 156744579/16807*x^6 - 2041518032/84035*x^5 + 1259355443184/14706125*x^4 + 3094420220918/14706125*x^3 + 123060442043827/367653125*x^2 + 82963044474852/367653125*x + 211679465261391/1838265625
sage: f = P.numerical_approx(500)[1].algdep(12); f / f.leading_coefficient()
x^12 + 6*x^11 + 90089/1715*x^10 + 71224/343*x^9 + 52563964/588245*x^8 - 483814934/588245*x^7 - 156744579/16807*x^6 - 2041518032/84035*x^5 + 1259355443184/14706125*x^4 + 3094420220918/14706125*x^3 + 123060442043827/367653125*x^2 + 82963044474852/367653125*x + 211679465261391/1838265625

sage: E = EllipticCurve('5077a')
sage: P = E.heegner_point(-7)
sage: P.point_exact(prec=100)
(0 : 1 : 0)
quadratic_form()

Return the integral primitive positive definite binary quadratic form associated to this Heegner point.

EXAMPLES:

sage: EllipticCurve('389a').heegner_point(-7, 5).quadratic_form()
389*x^2 + 147*x*y + 14*y^2

sage: P = EllipticCurve('389a').heegner_point(-7, 5, (778,925,275)); P
Heegner point of discriminant -7 and conductor 5 on elliptic curve of conductor 389
sage: P.quadratic_form()
778*x^2 + 925*x*y + 275*y^2
satisfies_kolyvagin_hypothesis(n=None)

Return True if this Heegner point and \(n\) satisfy the Kolyvagin hypothesis, i.e., that each prime dividing the conductor \(c\) of self is inert in K and coprime to \(ND\). Moreover, if \(n\) is not None, also check that for each prime \(p\) dividing \(c\) we have that \(n | \gcd(a_p(E), p+1)\).

INPUT:

\(n\) – positive integer

EXAMPLES:

sage: EllipticCurve('389a').heegner_point(-7).satisfies_kolyvagin_hypothesis()
True
sage: EllipticCurve('389a').heegner_point(-7,5).satisfies_kolyvagin_hypothesis()
True
sage: EllipticCurve('389a').heegner_point(-7,11).satisfies_kolyvagin_hypothesis()
False
tau()

Return \(\tau\) in the upper half plane that maps via the modular parametrization to this Heegner point.

EXAMPLES:

sage: E = EllipticCurve('389a'); P = E.heegner_point(-7, 5)
sage: P.tau()
5/778*sqrt_minus_7 - 147/778
x_poly_exact(prec=53, algorithm='lll')

Return irreducible polynomial over the rational numbers satisfied by the \(x\) coordinate of this Heegner point. A ValueError is raised if the precision is clearly insignificant to define a point on the curve.

Warning

It is in theory possible for this function to not raise a ValueError, find a polynomial, but via some very unlikely coincidence that point is not actually this Heegner point.

INPUT:

  • prec – integer (default: 53)

  • algorithm – ‘conjugates’ or ‘lll’ (default); if

    ‘conjugates’, compute numerically all the conjugates y[i] of the Heegner point and construct the characteristic polynomial as the product \(f(X)=(X-y[i])\). If ‘lll’, compute only one of the conjugates y[0], then uses the LLL algorithm to guess \(f(X)\).

EXAMPLES:

We compute some \(x\)-coordinate polynomials of some conductor 1 Heegner points:

sage: E = EllipticCurve('37a')
sage: v = E.heegner_discriminants_list(10)
sage: [E.heegner_point(D).x_poly_exact() for D in v]
[x, x, x^2 + 2, x^5 - x^4 + x^3 + x^2 - 2*x + 1, x - 6, x^7 - 2*x^6 + 9*x^5 - 10*x^4 - x^3 + 8*x^2 - 5*x + 1, x^3 + 5*x^2 + 10*x + 4, x^4 - 10*x^3 + 10*x^2 + 12*x - 12, x^8 - 5*x^7 + 7*x^6 + 13*x^5 - 10*x^4 - 4*x^3 + x^2 - 5*x + 7, x^6 - 2*x^5 + 11*x^4 - 24*x^3 + 30*x^2 - 16*x + 4]

We compute \(x\)-coordinate polynomials for some Heegner points of conductor bigger than 1 on a rank 2 curve:

sage: E = EllipticCurve('389a'); P = E.heegner_point(-7, 5); P
Heegner point of discriminant -7 and conductor 5 on elliptic curve of conductor 389
sage: P.x_poly_exact()
Traceback (most recent call last):
...
ValueError: insufficient precision to determine Heegner point (fails discriminant test)
sage: P.x_poly_exact(75)
x^6 + 10/7*x^5 - 867/49*x^4 - 76/245*x^3 + 3148/35*x^2 - 25944/245*x + 48771/1225
sage: E.heegner_point(-7,11).x_poly_exact(300)
x^10 + 282527/52441*x^9 + 27049007420/2750058481*x^8 - 22058564794/2750058481*x^7 - 140054237301/2750058481*x^6 + 696429998952/30250643291*x^5 + 2791387923058/30250643291*x^4 - 3148473886134/30250643291*x^3 + 1359454055022/30250643291*x^2 - 250620385365/30250643291*x + 181599685425/332757076201

Here we compute a Heegner point of conductor 5 on a rank 3 curve:

sage: E = EllipticCurve('5077a'); P = E.heegner_point(-7,5); P
Heegner point of discriminant -7 and conductor 5 on elliptic curve of conductor 5077
sage: P.x_poly_exact(300)
x^6 + 1108754853727159228/72351048803252547*x^5 + 88875505551184048168/1953478317687818769*x^4 - 2216200271166098662132/3255797196146364615*x^3 + 14941627504168839449851/9767391588439093845*x^2 - 3456417460183342963918/3255797196146364615*x + 1306572835857500500459/5426328660243941025
class sage.schemes.elliptic_curves.heegner.HeegnerPointOnX0N(N, D, c=1, f=None, check=True)

Bases: sage.schemes.elliptic_curves.heegner.HeegnerPoint

A Heegner point as a point on the modular curve \(X_0(N)\), which we view as the upper half plane modulo the action of \(\Gamma_0(N)\).

EXAMPLES:

sage: x = heegner_point(37,-7,5); x
Heegner point 5/74*sqrt(-7) - 11/74 of discriminant -7 and conductor 5 on X_0(37)
sage: type(x)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPointOnX0N'>
sage: x.level()
37
sage: x.conductor()
5
sage: x.discriminant()
-7
sage: x.quadratic_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
sage: x.quadratic_form()
37*x^2 + 11*x*y + 2*y^2
sage: x.quadratic_order()
Order in Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
sage: x.tau()
5/74*sqrt_minus_7 - 11/74
sage: loads(dumps(x)) == x
True
atkin_lehner_act(Q=None)

Given an integer Q dividing the level N such that \(\gcd(Q, N/Q) = 1\), returns the image of this Heegner point under the Atkin-Lehner operator \(W_Q\).

INPUT:

  • \(Q\) – positive divisor of \(N\); if not given, default to \(N\)

EXAMPLES:

sage: x = heegner_point(389,-7,5)
sage: x.atkin_lehner_act()
Heegner point 5/199168*sqrt(-7) - 631/199168 of discriminant -7 and conductor 5 on X_0(389)

sage: x = heegner_point(45,D=-11,c=1); x
Heegner point 1/90*sqrt(-11) - 13/90 of discriminant -11 on X_0(45)
sage: x.atkin_lehner_act(5)
Heegner point 1/90*sqrt(-11) + 23/90 of discriminant -11 on X_0(45)
sage: y = x.atkin_lehner_act(9); y
Heegner point 1/90*sqrt(-11) - 23/90 of discriminant -11 on X_0(45)
sage: z = y.atkin_lehner_act(9); z
Heegner point 1/90*sqrt(-11) - 13/90 of discriminant -11 on X_0(45)
sage: z == x
True
galois_orbit_over_K()

Return the \(Gal(K_c/K)\)-orbit of this Heegner point.

EXAMPLES:

sage: x = heegner_point(389,-7,3); x
Heegner point 3/778*sqrt(-7) - 223/778 of discriminant -7 and conductor 3 on X_0(389)
sage: x.galois_orbit_over_K()
[Heegner point 3/778*sqrt(-7) - 223/778 of discriminant -7 and conductor 3 on X_0(389), Heegner point 3/1556*sqrt(-7) - 223/1556 of discriminant -7 and conductor 3 on X_0(389), Heegner point 3/1556*sqrt(-7) - 1001/1556 of discriminant -7 and conductor 3 on X_0(389), Heegner point 3/3112*sqrt(-7) - 223/3112 of discriminant -7 and conductor 3 on X_0(389)]
map_to_curve(E)

Return the image of this Heegner point on the elliptic curve \(E\), which must also have conductor \(N\), where \(N\) is the level of self.

EXAMPLES:

sage: x = heegner_point(389,-7,5); x
Heegner point 5/778*sqrt(-7) - 147/778 of discriminant -7 and conductor 5 on X_0(389)
sage: y = x.map_to_curve(EllipticCurve('389a')); y
Heegner point of discriminant -7 and conductor 5 on elliptic curve of conductor 389
sage: y.curve().cremona_label()
'389a1'
sage: y.heegner_point_on_X0N()
Heegner point 5/778*sqrt(-7) - 147/778 of discriminant -7 and conductor 5 on X_0(389)

You can also directly apply the modular parametrization of the elliptic curve:

sage: x = heegner_point(37,-7); x
Heegner point 1/74*sqrt(-7) - 17/74 of discriminant -7 on X_0(37)
sage: E = EllipticCurve('37a'); phi = E.modular_parametrization()
sage: phi(x)
Heegner point of discriminant -7 on elliptic curve of conductor 37
plot(**kwds)

Draw a point at \((x,y)\) where this Heegner point is represented by the point \(\tau = x + i y\) in the upper half plane.

The kwds get passed onto the point plotting command.

EXAMPLES:

sage: heegner_point(389,-7,1).plot(pointsize=50)
quadratic_form()

Return the integral primitive positive-definite binary quadratic form associated to this Heegner point.

EXAMPLES:

sage: heegner_point(389,-7,5).quadratic_form()
389*x^2 + 147*x*y + 14*y^2
reduced_quadratic_form()

Return reduced binary quadratic corresponding to this Heegner point.

EXAMPLES:

sage: x = heegner_point(389,-7,5)
sage: x.quadratic_form()
389*x^2 + 147*x*y + 14*y^2
sage: x.reduced_quadratic_form()
4*x^2 - x*y + 11*y^2
tau()

Return an element tau in the upper half plane that corresponds to this particular Heegner point (actually, tau is in the quadratic imagqinary field K associated to this Heegner point).

EXAMPLES:

sage: x = heegner_point(37,-7,5); tau = x.tau(); tau
5/74*sqrt_minus_7 - 11/74
sage: 37 * tau.minpoly()
37*x^2 + 11*x + 2
sage: x.quadratic_form()
37*x^2 + 11*x*y + 2*y^2
class sage.schemes.elliptic_curves.heegner.HeegnerPoints(N)

Bases: sage.structure.sage_object.SageObject

The set of Heegner points with given parameters.

EXAMPLES:

sage: H = heegner_points(389); H
Set of all Heegner points on X_0(389)
sage: type(H)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPoints_level'>
sage: isinstance(H, sage.schemes.elliptic_curves.heegner.HeegnerPoints)
True
level()

Return the level \(N\) of the modular curve \(X_0(N)\).

EXAMPLES:

sage: heegner_points(389).level()
389
class sage.schemes.elliptic_curves.heegner.HeegnerPoints_level(N)

Bases: sage.schemes.elliptic_curves.heegner.HeegnerPoints

Return the infinite set of all Heegner points on \(X_0(N)\) for all quadratic imaginary fields.

EXAMPLES:

sage: H = heegner_points(11); H
Set of all Heegner points on X_0(11)
sage: type(H)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPoints_level'>
sage: loads(dumps(H)) == H
True
discriminants(n=10, weak=False)

Return the first \(n\) quadratic imaginary discriminants that satisfy the Heegner hypothesis for \(N\).

INPUTS:

  • \(n\) – nonnegative integer
  • weak – bool (default: False); if True only require weak Heegner hypothesis, which is the same as usual but without the condition that \(\gcd(D,N)=1\).

EXAMPLES:

sage: X = heegner_points(37)
sage: X.discriminants(5)
[-7, -11, -40, -47, -67]

The default is 10:

sage: X.discriminants()
[-7, -11, -40, -47, -67, -71, -83, -84, -95, -104]
sage: X.discriminants(15)
[-7, -11, -40, -47, -67, -71, -83, -84, -95, -104, -107, -115, -120, -123, -127]

The discriminant -111 satisfies only the weak Heegner hypothesis, since it is divisible by 37:

sage: X.discriminants(15,weak=True)
[-7, -11, -40, -47, -67, -71, -83, -84, -95, -104, -107, -111, -115, -120, -123]
reduce_mod(ell)

Return object that allows for computation with Heegner points of level \(N\) modulo the prime \(\ell\), represented using quaternion algebras.

INPUT:

  • \(\ell\) – prime

EXAMPLES:

sage: heegner_points(389).reduce_mod(7).quaternion_algebra()
Quaternion Algebra (-1, -7) with base ring Rational Field
class sage.schemes.elliptic_curves.heegner.HeegnerPoints_level_disc(N, D)

Bases: sage.schemes.elliptic_curves.heegner.HeegnerPoints

Set of Heegner points of given level and all conductors associated to a quadratic imaginary field.

EXAMPLES:

sage: H = heegner_points(389,-7); H
Set of all Heegner points on X_0(389) associated to QQ[sqrt(-7)]
sage: type(H)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPoints_level_disc'>
sage: H._repr_()
'Set of all Heegner points on X_0(389) associated to QQ[sqrt(-7)]'
sage: H.discriminant()
-7
sage: H.quadratic_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
sage: H.kolyvagin_conductors()
[1, 3, 5, 13, 15, 17, 19, 31, 39, 41]

sage: loads(dumps(H)) == H
True
discriminant()

Return the discriminant of the quadratic imaginary extension \(K\).

EXAMPLES:

sage: heegner_points(389,-7).discriminant()
-7
kolyvagin_conductors(r=None, n=10, E=None, m=None)

Return the first \(n\) conductors that are squarefree products of distinct primes inert in the quadratic imaginary field \(K = \QQ(\sqrt{D})\). If \(r\) is specified, return only conductors that are a product of \(r\) distinct primes all inert in \(K\). If \(r = 0\), always return the list [1], no matter what.

If the optional elliptic curve \(E\) and integer \(m\) are given, then only include conductors \(c\) such that for each prime divisor \(p\) of \(c\) we have \(m \mid \gcd(a_p(E), p+1)\).

INPUT:

  • \(r\) – (default: None) nonnegative integer or None
  • \(n\) – positive integer
  • \(E\) – an elliptic curve
  • \(m\) – a positive integer

EXAMPLES:

sage: H = heegner_points(389,-7)
sage: H.kolyvagin_conductors(0)
[1]
sage: H.kolyvagin_conductors(1)
[3, 5, 13, 17, 19, 31, 41, 47, 59, 61]
sage: H.kolyvagin_conductors(1,15)
[3, 5, 13, 17, 19, 31, 41, 47, 59, 61, 73, 83, 89, 97, 101]
sage: H.kolyvagin_conductors(1,5)
[3, 5, 13, 17, 19]
sage: H.kolyvagin_conductors(1,5,EllipticCurve('389a'),3)
[5, 17, 41, 59, 83]
sage: H.kolyvagin_conductors(2,5,EllipticCurve('389a'),3)
[85, 205, 295, 415, 697]
quadratic_field()

Return the quadratic imaginary field \(K = \QQ(\sqrt{D})\).

EXAMPLES:

sage: E = EllipticCurve('389a'); K = E.heegner_point(-7,5).ring_class_field()
sage: K.quadratic_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
class sage.schemes.elliptic_curves.heegner.HeegnerPoints_level_disc_cond(N, D, c=1)

Bases: sage.schemes.elliptic_curves.heegner.HeegnerPoints_level, sage.schemes.elliptic_curves.heegner.HeegnerPoints_level_disc

The set of Heegner points of given level, discriminant, and conductor.

EXAMPLES:

sage: H = heegner_points(389,-7,5); H
All Heegner points of conductor 5 on X_0(389) associated to QQ[sqrt(-7)]
sage: type(H)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPoints_level_disc_cond'>
sage: H.discriminant()
-7
sage: H.level()
389

sage: len(H.points())
12
sage: H.points()[0]
Heegner point 5/778*sqrt(-7) - 147/778 of discriminant -7 and conductor 5 on X_0(389)
sage: H.betas()
(147, 631)

sage: H.quadratic_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
sage: H.ring_class_field()
Ring class field extension of QQ[sqrt(-7)] of conductor 5

sage: H.kolyvagin_conductors()
[1, 3, 5, 13, 15, 17, 19, 31, 39, 41]
sage: H.satisfies_kolyvagin_hypothesis()
True

sage: H = heegner_points(389,-7,5)
sage: loads(dumps(H)) == H
True
betas()

Return the square roots of \(D c^2\) modulo \(4 N\) all reduced mod \(2 N\), without multiplicity.

EXAMPLES:

sage: X = heegner_points(45,-11,1); X
All Heegner points of conductor 1 on X_0(45) associated to QQ[sqrt(-11)]
sage: [x.quadratic_form() for x in X]
[45*x^2 + 13*x*y + y^2,
 45*x^2 + 23*x*y + 3*y^2,
 45*x^2 + 67*x*y + 25*y^2,
 45*x^2 + 77*x*y + 33*y^2]
sage: X.betas()
(13, 23, 67, 77)
sage: X.points(13)
(Heegner point 1/90*sqrt(-11) - 13/90 of discriminant -11 on X_0(45),)
sage: [x.quadratic_form() for x in X.points(13)]
[45*x^2 + 13*x*y + y^2]
conductor()

Return the level of the conductor.

EXAMPLES:

sage: heegner_points(389,-7,5).conductor()
5
plot(*args, **kwds)

Returns plot of all the representatives in the upper half plane of the Heegner points in this set of Heegner points.

The inputs to this function get passed onto the point command.

EXAMPLES:

sage: heegner_points(389,-7,5).plot(pointsize=50, rgbcolor='red')
sage: heegner_points(53,-7,15).plot(pointsize=50, rgbcolor='purple')
points(beta=None)

Return the Heegner points in self. If \(\beta\) is given, return only those Heegner points with given \(\beta\), i.e., whose quadratic form has \(B\) congruent to \(\beta\) modulo \(2 N\).

Use self.beta() to get a list of betas.

EXAMPLES:

sage: H = heegner_points(389,-7,5); H
All Heegner points of conductor 5 on X_0(389) associated to QQ[sqrt(-7)]
sage: H.points()
(Heegner point 5/778*sqrt(-7) - 147/778 of discriminant -7 and conductor 5 on X_0(389), ..., Heegner point 5/5446*sqrt(-7) - 757/778 of discriminant -7 and conductor 5 on X_0(389))
sage: H.betas()
(147, 631)
sage: [x.tau() for x in H.points(147)]
[5/778*sqrt_minus_7 - 147/778, 5/1556*sqrt_minus_7 - 147/1556, 5/1556*sqrt_minus_7 - 925/1556, 5/3112*sqrt_minus_7 - 1703/3112, 5/3112*sqrt_minus_7 - 2481/3112, 5/5446*sqrt_minus_7 - 21/778]

sage: [x.tau() for x in H.points(631)]
[5/778*sqrt_minus_7 - 631/778, 5/1556*sqrt_minus_7 - 631/1556, 5/1556*sqrt_minus_7 - 1409/1556, 5/3112*sqrt_minus_7 - 631/3112, 5/3112*sqrt_minus_7 - 1409/3112, 5/5446*sqrt_minus_7 - 757/778]

The result is cached and is a tuple (since it is immutable):

sage: H.points() is H.points()
True
sage: type(H.points())
<type 'tuple'>
ring_class_field()

Return the ring class field associated to this set of Heegner points. This is an extension \(K_c\) over \(K\), where \(K\) is the quadratic imaginary field and \(c\) the conductor associated to this Heegner point. This Heegner point is defined over \(K_c\) and the Galois group \(Gal(K_c/K)\) acts transitively on the Galois conjugates of this Heegner point.

EXAMPLES:

sage: heegner_points(389,-7,5).ring_class_field()
Ring class field extension of QQ[sqrt(-7)] of conductor 5
satisfies_kolyvagin_hypothesis()

Return True if self satisfies the Kolyvagin hypothesis, i.e., that each prime dividing the conductor \(c\) of self is inert in \(K\) and coprime to \(ND\).

EXAMPLES:

The prime 5 is inert, but the prime 11 is not:

sage: heegner_points(389,-7,5).satisfies_kolyvagin_hypothesis()
True
sage: heegner_points(389,-7,11).satisfies_kolyvagin_hypothesis()
False
class sage.schemes.elliptic_curves.heegner.HeegnerQuatAlg(level, ell)

Bases: sage.structure.sage_object.SageObject

Heegner points viewed as supersingular points on the modular curve \(X_0(N)/\mathbf{F}_{\ell}\).

EXAMPLES:

sage: H = heegner_points(11).reduce_mod(13); H
Heegner points on X_0(11) over F_13
sage: type(H)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerQuatAlg'>
sage: loads(dumps(H)) == H
True
brandt_module()

Return the Brandt module of right ideal classes that we used to represent the set of supersingular points on the modular curve.

EXAMPLES:

sage: heegner_points(11).reduce_mod(3).brandt_module()
Brandt module of dimension 2 of level 3*11 of weight 2 over Rational Field
cyclic_subideal_p1(I, c)

Compute dictionary mapping 2-tuples that defined normalized elements of \(P^1(\ZZ/c\ZZ)\)

INPUT:

  • \(I\) – right ideal of Eichler order or in quaternion algebra

  • \(c\) – square free integer (currently must be odd prime

    and coprime to level, discriminant, characteristic, etc.

OUTPUT:

  • dictionary mapping 2-tuples (u,v) to ideals

EXAMPLES:

sage: H = heegner_points(11).reduce_mod(7)
sage: I = H.brandt_module().right_ideals()[0]
sage: sorted(H.cyclic_subideal_p1(I,3).iteritems())
[((0, 1),
  Fractional ideal (2 + 2*j + 32*k, 2*i + 8*j + 82*k, 12*j + 60*k, 132*k)),
 ((1, 0),
  Fractional ideal (2 + 10*j + 28*k, 2*i + 4*j + 62*k, 12*j + 60*k, 132*k)),
 ((1, 1),
  Fractional ideal (2 + 2*j + 76*k, 2*i + 4*j + 106*k, 12*j + 60*k, 132*k)),
 ((1, 2),
  Fractional ideal (2 + 10*j + 116*k, 2*i + 8*j + 38*k, 12*j + 60*k, 132*k))]
sage: len(H.cyclic_subideal_p1(I,17))
18
ell()

Return the prime \(\ell\) modulo which we are working.

EXAMPLES:

sage: heegner_points(11).reduce_mod(3).ell()
3
galois_group_over_hilbert_class_field(D, c)

Return the Galois group of the extension of ring class fields \(K_c\) over the Hilbert class field \(K_{1}\) of the quadratic imaginary field of discriminant \(D\).

INPUT:

  • \(D\) – fundamental discriminant
  • \(c\) – conductor (square-free integer)

EXAMPLES:

sage: N = 37; D = -7; ell = 17; c = 41; p = 3
sage: H = heegner_points(N).reduce_mod(ell)
sage: H.galois_group_over_hilbert_class_field(D, c)
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 41 over Hilbert class field of QQ[sqrt(-7)]
galois_group_over_quadratic_field(D, c)

Return the Galois group of the extension of ring class fields \(K_c\) over the quadratic imaginary field \(K\) of discriminant \(D\).

INPUT:

  • \(D\) – fundamental discriminant
  • \(c\) – conductor (square-free integer)

EXAMPLES:

sage: N = 37; D = -7; ell = 17; c = 41; p = 3
sage: H = heegner_points(N).reduce_mod(ell)
sage: H.galois_group_over_quadratic_field(D, c)
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 41 over Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
heegner_conductors(D, n=5)

Return the first \(n\) negative fundamental discriminants coprime to \(N\ell\) such that \(\ell\) is inert in the corresponding quadratic imaginary field and that field satisfies the Heegner hypothesis.

INPUT:

  • \(D\) – negative integer; a fundamental Heegner discriminant
  • \(n\) – positive integer (default: 5)

OUTPUT:

  • list

EXAMPLES:

sage: H = heegner_points(11).reduce_mod(3)
sage: H.heegner_conductors(-7)
[1, 2, 4, 5, 8]
sage: H.heegner_conductors(-7, 10)
[1, 2, 4, 5, 8, 10, 13, 16, 17, 19]
heegner_discriminants(n=5)

Return the first \(n\) negative fundamental discriminants coprime to \(N\ell\) such that \(\ell\) is inert in the corresponding quadratic imaginary field and that field satisfies the Heegner hypothesis, and \(N\) is the level.

INPUT:

  • \(n\) – positive integer (default: 5)

OUTPUT:

  • list

EXAMPLES:

sage: H = heegner_points(11).reduce_mod(3)
sage: H.heegner_discriminants()
[-7, -19, -40, -43, -52]
sage: H.heegner_discriminants(10)
[-7, -19, -40, -43, -52, -79, -127, -139, -151, -184]
heegner_divisor(D, c=1)

Return Heegner divisor as an element of the Brandt module corresponding to the discriminant \(D\) and conductor \(c\), which both must be coprime to \(N\ell\).

More precisely, we compute the sum of the reductions of the \(\textrm{Gal}(K_1/K)\)-conjugates of each choice of \(y_1\), where the choice comes from choosing the ideal \(\mathcal{N}\). Then we apply the Hecke operator \(T_c\) to this sum.

INPUT:

  • \(D\) – discriminant (negative integer)
  • \(c\) – conductor (positive integer)

OUTPUT:

  • Brandt module element

EXAMPLES:

sage: H = heegner_points(11).reduce_mod(7)
sage: H.heegner_discriminants()
[-8, -39, -43, -51, -79]
sage: H.heegner_divisor(-8)
(1, 0, 0, 1, 0, 0)
sage: H.heegner_divisor(-39)
(1, 2, 2, 1, 2, 0)
sage: H.heegner_divisor(-43)
(1, 0, 0, 1, 0, 0)
sage: H.heegner_divisor(-51)
(1, 0, 0, 1, 0, 2)
sage: H.heegner_divisor(-79)
(3, 2, 2, 3, 0, 0)

sage: sum(H.heegner_divisor(-39).element())
8
sage: QuadraticField(-39,'a').class_number()
4
kolyvagin_cyclic_subideals(I, p, alpha_quaternion)

Return list of pairs \((J, n)\) where \(J\) runs through the cyclic subideals of \(I\) of index \((\ZZ/p\ZZ)^2\), and \(J \sim \alpha^n(J_0)\) for some fixed choice of cyclic subideal \(J_0\).

INPUT:

  • \(I\) – right ideal of the quaternion algebra

  • \(p\) – prime number

  • alpha_quaternion – image in the quaternion algebra

    of generator \(\alpha\) for \((\mathcal{O}_K / c\mathcal{O}_K)^* / (\ZZ/c\ZZ)^*\).

OUTPUT:

  • list of 2-tuples

EXAMPLES:

sage: N = 37; D = -7; ell = 17; c=5
sage: H = heegner_points(N).reduce_mod(ell)
sage: B = H.brandt_module(); I = B.right_ideals()[32]
sage: f = H.optimal_embeddings(D, 1, I.left_order())[0]
sage: g = H.kolyvagin_generators(f.domain().number_field(), c)
sage: alpha_quaternion = f(g[0]); alpha_quaternion
1 - 5/128*i - 77/192*j + 137/384*k
sage: H.kolyvagin_cyclic_subideals(I, 5, alpha_quaternion)
[(Fractional ideal (2 + 874/3*j + 128356/3*k, 2*i + 932/3*j + 198806/3*k, 2560/3*j + 33280/3*k, 94720*k), 0), (Fractional ideal (2 + 462*j + 82892*k, 2*i + 932/3*j + 141974/3*k, 2560/3*j + 33280/3*k, 94720*k), 1), (Fractional ideal (2 + 2410/3*j + 261988/3*k, 2*i + 652*j + 89650*k, 2560/3*j + 33280/3*k, 94720*k), 2), (Fractional ideal (2 + 2410/3*j + 91492/3*k, 2*i + 1444/3*j + 148630/3*k, 2560/3*j + 33280/3*k, 94720*k), 3), (Fractional ideal (2 + 874/3*j + 71524/3*k, 2*i + 2468/3*j + 275606/3*k, 2560/3*j + 33280/3*k, 94720*k), 4), (Fractional ideal (2 + 462*j + 63948*k, 2*i + 2468/3*j + 218774/3*k, 2560/3*j + 33280/3*k, 94720*k), 5)]
kolyvagin_generator(K, p)

Return element in \(K\) that maps to the multiplicative generator for the quotient group

\((\mathcal{O}_K / p \mathcal{O}_K)^* / (\ZZ/p\ZZ)^*\)

of the form \(\sqrt{D}+n\) with \(n\geq 1\) minimal.

INPUT:

  • \(K\) – quadratic imaginary field
  • \(p\) – inert prime

EXAMPLES:

sage: N = 37; D = -7; ell = 17; p=5
sage: H = heegner_points(N).reduce_mod(ell)
sage: B = H.brandt_module(); I = B.right_ideals()[32]
sage: f = H.optimal_embeddings(D, 1, I.left_order())[0]
sage: H.kolyvagin_generator(f.domain().number_field(), 5)
a + 1

This function requires that p be prime, but kolyvagin_generators works in general:

sage: H.kolyvagin_generator(f.domain().number_field(), 5*17)
Traceback (most recent call last):
...
NotImplementedError: p must be prime
sage: H.kolyvagin_generators(f.domain().number_field(), 5*17)
[-34*a + 1, 35*a + 106]
kolyvagin_generators(K, c)

Return elements in \(\mathcal{O}_K\) that map to multiplicative generators for the factors of the quotient group

\((\mathcal{O}_K / c \mathcal{O}_K)^* / (\ZZ/c\ZZ)^*\)

corresponding to the prime divisors of c. Each generator is of the form \(\sqrt{D}+n\) with \(n\geq 1\) minimal.

INPUT:

  • \(K\) – quadratic imaginary field
  • \(c\) – square free product of inert prime

EXAMPLES:

sage: N = 37; D = -7; ell = 17; p=5
sage: H = heegner_points(N).reduce_mod(ell)
sage: B = H.brandt_module(); I = B.right_ideals()[32]
sage: f = H.optimal_embeddings(D, 1, I.left_order())[0]
sage: H.kolyvagin_generators(f.domain().number_field(), 5*17)
[-34*a + 1, 35*a + 106]
kolyvagin_point_on_curve(D, c, E, p, bound=10)

Compute image of the Kolyvagin divisor \(P_c\) in \(E(\GF{\ell^2}) / p E(\GF{\ell^2})\). Note that this image is by definition only well defined up to scalars. However, doing multiple computations will always yield the same result, and working modulo different \(\ell\) is compatible (since we always chose the same generator for \(\textrm{Gal}(K_c/K_1)\)).

INPUT:

  • \(D\) – fundamental negative discriminant

  • \(c\) – conductor

  • \(E\) – elliptic curve of conductor the level of self

  • \(p\) – odd prime number such that we consider image in

    \(E(\GF{\ell^2}) / p E(\GF{\ell^2})\)

  • bound – integer (default: 10)

EXAMPLES:

sage: N = 37; D = -7; ell = 17; c = 41; p = 3
sage: H = heegner_points(N).reduce_mod(ell)
sage: H.kolyvagin_point_on_curve(D, c, EllipticCurve('37a'), p)
[1, 1]
kolyvagin_sigma_operator(D, c, r, bound=None)

Return the action of the Kolyvagin sigma operator on the \(r\)-th basis vector.

INPUT:

  • \(D\) – fundamental discriminant
  • \(c\) – conductor (square-free integer, need not be prime)
  • \(r\) – nonnegative integer
  • bound – (default: None), if given, controls precision of computation of theta series, which could impact performance, but does not impact correctness

EXAMPLES:

We first try to verify Kolyvagin’s conjecture for a rank 2 curve by working modulo 5, but we are unlucky with \(c=17\):

sage: N = 389; D = -7; ell = 5; c = 17; q = 3
sage: H = heegner_points(N).reduce_mod(ell)
sage: E = EllipticCurve('389a')
sage: V = H.modp_dual_elliptic_curve_factor(E, q, 5)  # long time (4s on sage.math, 2012)
sage: k118 = H.kolyvagin_sigma_operator(D, c, 118)
sage: k104 = H.kolyvagin_sigma_operator(D, c, 104)
sage: [b.dot_product(k104.element().change_ring(GF(3))) for b in V.basis()]  # long time
[0, 0]
sage: [b.dot_product(k118.element().change_ring(GF(3))) for b in V.basis()]  # long time
[0, 0]

Next we try again with \(c=41\) and this does work, in that we get something nonzero, when dotting with V:

sage: c = 41
sage: k118 = H.kolyvagin_sigma_operator(D, c, 118)
sage: k104 = H.kolyvagin_sigma_operator(D, c, 104)
sage: [b.dot_product(k118.element().change_ring(GF(3))) for b in V.basis()]  # long time
[1, 0]
sage: [b.dot_product(k104.element().change_ring(GF(3))) for b in V.basis()]  # long time
[2, 0]

By the way, the above is the first ever provable verification of Kolyvagin’s conjecture for any curve of rank at least 2.

Another example, but where the curve has rank 1:

sage: N = 37; D = -7; ell = 17; c = 41; q = 3
sage: H = heegner_points(N).reduce_mod(ell)
sage: H.heegner_divisor(D,1).element().nonzero_positions()
[32, 51]
sage: k32 = H.kolyvagin_sigma_operator(D, c, 32); k32
(63, 68, 47, 47, 31, 52, 37, 0, 0, 47, 3, 31, 47, 7, 21, 26, 19, 10, 0, 0, 11, 28, 41, 2, 47, 25, 0, 0, 36, 0, 33, 0, 0, 0, 40, 6, 14, 22, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
sage: k51 = H.kolyvagin_sigma_operator(D, c, 51); k51
(5, 13, 0, 0, 14, 0, 21, 0, 0, 0, 29, 0, 0, 45, 0, 6, 0, 40, 0, 61, 0, 0, 40, 32, 0, 9, 0, 0, 0, 0, 17, 0, 0, 0, 77, 40, 2, 10, 18, 0, 0, 61, 19, 45, 26, 80, 61, 35, 35, 19, 1, 0)
sage: V = H.modp_dual_elliptic_curve_factor(EllipticCurve('37a'), q, 5); V
Vector space of degree 52 and dimension 2 over Ring of integers modulo 3
Basis matrix:
2 x 52 dense matrix over Ring of integers modulo 3
sage: [b.dot_product(k32.element().change_ring(GF(q))) for b in V.basis()]
[1, 1]
sage: [b.dot_product(k51.element().change_ring(GF(q))) for b in V.basis()]
[1, 1]

An example with \(c\) a product of two primes:

sage: N = 389; D = -7; ell = 5; q = 3
sage: H = heegner_points(N).reduce_mod(ell)
sage: V = H.modp_dual_elliptic_curve_factor(EllipticCurve('389a'), q, 5)
sage: k = H.kolyvagin_sigma_operator(D, 17*41, 104)     # long time
sage: k                                                 # long time
(494, 472, 1923, 1067, ..., 102, 926)
sage: [b.dot_product(k.element().change_ring(GF(3))) for b in V.basis()]   # long time (but only because depends on something slow)
[0, 0]
left_orders()

Return the left orders associated to the representative right ideals in the Brandt module.

EXAMPLES:

sage: heegner_points(11).reduce_mod(3).left_orders()
[Order of Quaternion Algebra (-1, -3) with base ring Rational Field with basis (1/2 + 1/2*j + 7*k, 1/2*i + 13/2*k, j + 3*k, 11*k),
 Order of Quaternion Algebra (-1, -3) with base ring Rational Field with basis (1/2 + 1/2*j + 7*k, 1/4*i + 1/2*j + 63/4*k, j + 14*k, 22*k)]
level()

Return the level.

EXAMPLES:

sage: heegner_points(11).reduce_mod(3).level()
11
modp_dual_elliptic_curve_factor(E, p, bound=10)

Return the factor of the Brandt module space modulo \(p\) corresponding to the elliptic curve \(E\), cut out using Hecke operators up to bound.

INPUT:

  • \(E\) – elliptic curve of conductor equal to the level of self
  • \(p\) – prime number
  • \(bound\) – positive integer (default: 10)

EXAMPLES:

sage: N = 37; D = -7; ell = 17; c = 41; q = 3
sage: H = heegner_points(N).reduce_mod(ell)
sage: V = H.modp_dual_elliptic_curve_factor(EllipticCurve('37a'), q, 5); V
Vector space of degree 52 and dimension 2 over Ring of integers modulo 3
Basis matrix:
2 x 52 dense matrix over Ring of integers modulo 3
modp_splitting_data(p)

Return mod \(p\) splitting data for the quaternion algebra at the unramified prime \(p\). This is a pair of \(2\times 2\) matrices \(A\), \(B\) over the finite field \(\GF{p}\) such that if the quaternion algebra has generators \(i, j, k\), then the homomorphism sending \(i\) to \(A\) and \(j\) to \(B\) maps any maximal order homomorphically onto the ring of \(2\times 2\) matrices.

Because of how the homomorphism is defined, we must assume that the prime \(p\) is odd.

INPUT:

  • \(p\) – unramified odd prime

OUTPUT:

  • 2-tuple of matrices over finite field

EXAMPLES:

sage: H = heegner_points(11).reduce_mod(7)
sage: H.quaternion_algebra()
Quaternion Algebra (-1, -7) with base ring Rational Field
sage: I, J = H.modp_splitting_data(13)
sage: I
[ 0 12]
[ 1  0]
sage: J
[7 3]
[3 6]
sage: I^2
[12  0]
[ 0 12]
sage: J^2
[6 0]
[0 6]
sage: I*J == -J*I
True

The following is a good test because of the asserts in the code:

sage: v = [H.modp_splitting_data(p) for p in primes(13,200)]

Some edge cases:

sage: H.modp_splitting_data(11)
(
[ 0 10]  [6 1]
[ 1  0], [1 5]
)

Proper error handling:

sage: H.modp_splitting_data(7)
Traceback (most recent call last):
...
ValueError: p (=7) must be an unramified prime

sage: H.modp_splitting_data(2)
Traceback (most recent call last):
...
ValueError: p must be odd
modp_splitting_map(p)

Return (algebra) map from the (\(p\)-integral) quaternion algebra to the set of \(2\times 2\) matrices over \(\GF{p}\).

INPUT:

  • \(p\) – prime number

EXAMPLES:

sage: H = heegner_points(11).reduce_mod(7)
sage: f = H.modp_splitting_map(13)
sage: B = H.quaternion_algebra(); B
Quaternion Algebra (-1, -7) with base ring Rational Field
sage: i,j,k = H.quaternion_algebra().gens()
sage: a = 2+i-j+3*k; b = 7+2*i-4*j+k
sage: f(a*b)
[12  3]
[10  5]
sage: f(a)*f(b)
[12  3]
[10  5]
optimal_embeddings(D, c, R)

INPUT:

  • \(D\) – negative fundamental disriminant
  • \(c\) – integer coprime
  • \(R\) – Eichler order

EXAMPLES:

sage: H = heegner_points(11).reduce_mod(3)
sage: R = H.left_orders()[0]
sage: H.optimal_embeddings(-7, 1, R)
[Embedding sending sqrt(-7) to -i + j + k,
 Embedding sending sqrt(-7) to i - j - k]
sage: H.optimal_embeddings(-7, 2, R)
[Embedding sending 2*sqrt(-7) to -5*i + k,
 Embedding sending 2*sqrt(-7) to 5*i - k,
 Embedding sending 2*sqrt(-7) to -2*i + 2*j + 2*k,
 Embedding sending 2*sqrt(-7) to 2*i - 2*j - 2*k]
quadratic_field(D)

Return our fixed choice of quadratic imaginary field of discriminant \(D\).

INPUT:

  • \(D\) – fundamental discriminant

OUTPUT:

  • a quadratic number field

EXAMPLES:

sage: H = heegner_points(389).reduce_mod(5)
sage: H.quadratic_field(-7)
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
quaternion_algebra()

Return the rational quaternion algebra used to implement self.

EXAMPLES:

sage: heegner_points(389).reduce_mod(7).quaternion_algebra()
Quaternion Algebra (-1, -7) with base ring Rational Field
rational_kolyvagin_divisor(D, c)

Return the Kolyvagin divisor as an element of the Brandt module corresponding to the discriminant \(D\) and conductor \(c\), which both must be coprime to \(N\ell\).

INPUT:

  • \(D\) – discriminant (negative integer)
  • \(c\) – conductor (positive integer)

OUTPUT:

  • Brandt module element (or tuple of them)

EXAMPLES:

sage: N = 389; D = -7; ell = 5; c = 17; q = 3
sage: H = heegner_points(N).reduce_mod(ell)
sage: k = H.rational_kolyvagin_divisor(D, c); k  # long time (5s on sage.math, 2013)
(14, 16, 0, 0, ... 0, 0, 0)
sage: V = H.modp_dual_elliptic_curve_factor(EllipticCurve('389a'), q, 2)
sage: [b.dot_product(k.element().change_ring(GF(q))) for b in V.basis()]  # long time
[0, 0]
sage: k = H.rational_kolyvagin_divisor(D, 59)
sage: [b.dot_product(k.element().change_ring(GF(q))) for b in V.basis()]
[1, 0]
right_ideals()

Return representative right ideals in the Brandt module.

EXAMPLES:

sage: heegner_points(11).reduce_mod(3).right_ideals()
(Fractional ideal (2 + 2*j + 28*k, 2*i + 26*k, 4*j + 12*k, 44*k),
 Fractional ideal (2 + 2*j + 28*k, 2*i + 4*j + 38*k, 8*j + 24*k, 88*k))
satisfies_heegner_hypothesis(D, c=1)

The fundamental discriminant \(D\) must be coprime to \(N\ell\), and must define a quadratic imaginary field \(K\) in which \(\ell\) is inert. Also, all primes dividing \(N\) must split in \(K\), and \(c\) must be squarefree and coprime to \(ND\ell\).

INPUT:

  • \(D\) – negative integer
  • \(c\) – positive integer (default: 1)

OUTPUT:

  • bool

EXAMPLES:

sage: H = heegner_points(11).reduce_mod(7)
sage: H.satisfies_heegner_hypothesis(-5)
False
sage: H.satisfies_heegner_hypothesis(-7)
False
sage: H.satisfies_heegner_hypothesis(-8)
True
sage: [D for D in [-1,-2..-100] if H.satisfies_heegner_hypothesis(D)]
[-8, -39, -43, -51, -79, -95]
class sage.schemes.elliptic_curves.heegner.HeegnerQuatAlgEmbedding(D, c, R, beta)

Bases: sage.structure.sage_object.SageObject

The homomorphism \(\mathcal{O} \to R\), where \(\mathcal{O}\) is the order of conductor \(c\) in the quadratic field of discriminant \(D\), and \(R\) is an Eichler order in a quaternion algebra.

EXAMPLES:

sage: H = heegner_points(11).reduce_mod(3); R = H.left_orders()[0]
sage: f = H.optimal_embeddings(-7, 2, R)[0]; f
Embedding sending 2*sqrt(-7) to -5*i + k
sage: type(f)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerQuatAlgEmbedding'>
sage: loads(dumps(f)) == f
True
beta()

Return the element \(\beta\) in the quaternion algebra order that \(c\sqrt{D}\) maps to.

EXAMPLES:

sage: H = heegner_points(11).reduce_mod(3); R = H.left_orders()[0]
sage: H.optimal_embeddings(-7, 2, R)[0].beta()
-5*i + k
codomain()

Return the codomain of this embedding.

EXAMPLES:

sage: H = heegner_points(11).reduce_mod(3); R = H.left_orders()[0]
sage: H.optimal_embeddings(-7, 2, R)[0].codomain()
Order of Quaternion Algebra (-1, -3) with base ring Rational Field with basis (1/2 + 1/2*j + 7*k, 1/2*i + 13/2*k, j + 3*k, 11*k)
conjugate()

Return the conjugate of this embedding, which is also an embedding.

EXAMPLES:

sage: H = heegner_points(11).reduce_mod(3); R = H.left_orders()[0]
sage: f = H.optimal_embeddings(-7, 2, R)[0]
sage: f.conjugate()
Embedding sending 2*sqrt(-7) to 5*i - k
sage: f
Embedding sending 2*sqrt(-7) to -5*i + k
domain()

Return the domain of this embedding.

EXAMPLES:

sage: H = heegner_points(11).reduce_mod(3); R = H.left_orders()[0]
sage: H.optimal_embeddings(-7, 2, R)[0].domain()
Order in Number Field in a with defining polynomial x^2 + 7
domain_conductor()

Return the conductor of the domain.

EXAMPLES:

sage: H = heegner_points(11).reduce_mod(3); R = H.left_orders()[0]
sage: H.optimal_embeddings(-7, 2, R)[0].domain_conductor()
2
domain_gen()

Return the specific generator \(c \sqrt{D}\) for the domain order.

EXAMPLES:

sage: H = heegner_points(11).reduce_mod(3); R = H.left_orders()[0]
sage: f = H.optimal_embeddings(-7, 2, R)[0]
sage: f.domain_gen()
2*a
sage: f.domain_gen()^2
-28
matrix()

Return matrix over \(\QQ\) of this morphism, with respect to the basis 1, \(c\sqrt{D}\) of the domain and the basis \(1,i,j,k\) of the ambient rational quaternion algebra (which contains the domain).

EXAMPLES:

sage: H = heegner_points(11).reduce_mod(3); R = H.left_orders()[0]
sage: f = H.optimal_embeddings(-7, 1, R)[0]; f
Embedding sending sqrt(-7) to -i + j + k
sage: f.matrix()
[ 1  0  0  0]
[ 0 -1  1  1]
sage: f.conjugate().matrix()
[ 1  0  0  0]
[ 0  1 -1 -1]
class sage.schemes.elliptic_curves.heegner.KolyvaginCohomologyClass(kolyvagin_point, n)

Bases: sage.structure.sage_object.SageObject

A Kolyvagin cohomology class in \(H^1(K,E[n])\) or \(H^1(K,E)[n]\) attached to a Heegner point.

EXAMPLES:

sage: y = EllipticCurve('37a').heegner_point(-7)
sage: c = y.kolyvagin_cohomology_class(3); c
Kolyvagin cohomology class c(1) in H^1(K,E[3])
sage: type(c)
<class 'sage.schemes.elliptic_curves.heegner.KolyvaginCohomologyClassEn'>
sage: loads(dumps(c)) == c
True
sage: y.kolyvagin_cohomology_class(5)
Kolyvagin cohomology class c(1) in H^1(K,E[5])
conductor()

Return the integer \(c\) such that this cohomology class is associated to the Heegner point \(y_c\).

EXAMPLES:

sage: y = EllipticCurve('37a').heegner_point(-7,5)
sage: t = y.kolyvagin_cohomology_class()
sage: t.conductor()
5
heegner_point()

Return the Heegner point \(y_c\) to which this cohomology class is associated.

EXAMPLES:

sage: y = EllipticCurve('37a').heegner_point(-7,5)
sage: t = y.kolyvagin_cohomology_class()
sage: t.heegner_point()
Heegner point of discriminant -7 and conductor 5 on elliptic curve of conductor 37
kolyvagin_point()

Return the Kolyvagin point \(P_c\) to which this cohomology class is associated.

EXAMPLES:

sage: y = EllipticCurve('37a').heegner_point(-7,5)
sage: t = y.kolyvagin_cohomology_class()
sage: t.kolyvagin_point()
Kolyvagin point of discriminant -7 and conductor 5 on elliptic curve of conductor 37
n()

Return the integer \(n\) so that this is a cohomology class in \(H^1(K,E[n])\) or \(H^1(K,E)[n]\).

EXAMPLES:

sage: y = EllipticCurve('37a').heegner_point(-7)
sage: t = y.kolyvagin_cohomology_class(3); t
Kolyvagin cohomology class c(1) in H^1(K,E[3])
sage: t.n()
3
class sage.schemes.elliptic_curves.heegner.KolyvaginCohomologyClassEn(kolyvagin_point, n)

Bases: sage.schemes.elliptic_curves.heegner.KolyvaginCohomologyClass

EXAMPLES:

class sage.schemes.elliptic_curves.heegner.KolyvaginPoint(heegner_point)

Bases: sage.schemes.elliptic_curves.heegner.HeegnerPoint

A Kolyvagin point.

EXAMPLES:

We create a few Kolyvagin points:

sage: EllipticCurve('11a1').kolyvagin_point(-7)
Kolyvagin point of discriminant -7 on elliptic curve of conductor 11
sage: EllipticCurve('37a1').kolyvagin_point(-7)
Kolyvagin point of discriminant -7 on elliptic curve of conductor 37
sage: EllipticCurve('37a1').kolyvagin_point(-67)
Kolyvagin point of discriminant -67 on elliptic curve of conductor 37
sage: EllipticCurve('389a1').kolyvagin_point(-7, 5)
Kolyvagin point of discriminant -7 and conductor 5 on elliptic curve of conductor 389

One can also associated a Kolyvagin point to a Heegner point:

sage: y = EllipticCurve('37a1').heegner_point(-7); y
Heegner point of discriminant -7 on elliptic curve of conductor 37
sage: y.kolyvagin_point()
Kolyvagin point of discriminant -7 on elliptic curve of conductor 37

TESTS:

sage: y = EllipticCurve('37a1').heegner_point(-7)
sage: type(y)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPointOnEllipticCurve'>
sage: loads(dumps(y)) == y
True
curve()

Return the elliptic curve over \(\QQ\) on which this Kolyvagin point sits.

EXAMPLES:

sage: E = EllipticCurve('37a1'); P = E.kolyvagin_point(-67, 3)
sage: P.curve()
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
heegner_point()

This Kolyvagin point \(P_c\) is associated to some Heegner point \(y_c\) via Kolyvagin’s construction. This function returns that point \(y_c\).

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: P = E.kolyvagin_point(-67); P
Kolyvagin point of discriminant -67 on elliptic curve of conductor 37
sage: y = P.heegner_point(); y
Heegner point of discriminant -67 on elliptic curve of conductor 37
sage: y.kolyvagin_point() is P
True
index(*args, **kwds)

Return index of this Kolyvagin point in the full group of \(K_c\) rational points on \(E\).

When the conductor is 1, this is computed numerically using the Gross-Zagier formula and explicit point search, and it may be off by \(2\). See the documentation for E.heegner_index, where \(E\) is the curve attached to self.

EXAMPLES:

sage: E = EllipticCurve('37a1'); P = E.kolyvagin_point(-67); P.index()
6
kolyvagin_cohomology_class(n=None)

INPUT:

  • \(n\) – positive integer that divides the gcd of \(a_p\) and \(p+1\) for all \(p\) dividing the conductor. If \(n\) is None, choose the largest valid \(n\).

EXAMPLES:

sage: y = EllipticCurve('389a').heegner_point(-7,5)
sage: P = y.kolyvagin_point()
sage: P.kolyvagin_cohomology_class(3)
Kolyvagin cohomology class c(5) in H^1(K,E[3])

sage: y = EllipticCurve('37a').heegner_point(-7,5).kolyvagin_point()
sage: y.kolyvagin_cohomology_class()
Kolyvagin cohomology class c(5) in H^1(K,E[2])
mod(p, prec=53)

Return the trace of the reduction \(Q\) modulo a prime over \(p\) of this Kolyvagin point as an element of \(E(\GF{p})\), where \(p\) is any prime that is inert in \(K\) that is coprime to \(NDc\).

The point \(Q\) is only well defined up to an element of \((p+1) E(\GF{p})\), i.e., it gives a well defined element of the abelian group \(E(\GF{p}) / (p+1) E(\GF{p})\).

See [SteinToward], Proposition 5.4 for a proof of the above well-definedness assertion.

EXAMPLES:

A Kolyvagin point on a rank 1 curve:

sage: E = EllipticCurve('37a1'); P = E.kolyvagin_point(-67)
sage: P.mod(2)
(1 : 1 : 1)
sage: P.mod(3)
(1 : 0 : 1)
sage: P.mod(5)
(2 : 2 : 1)
sage: P.mod(7)
(6 : 0 : 1)
sage: P.trace_to_real_numerical()
(1.61355529131986 : -2.18446840788880 : 1.00000000000000)
sage: P._trace_exact_conductor_1()  # the actual point we're reducing
(1357/841 : -53277/24389 : 1)
sage: (P._trace_exact_conductor_1().height() / E.regulator()).sqrt()
12.0000000000000

Here the Kolyvagin point is a torsion point (since \(E\) has rank 1), and we reduce it modulo several primes.:

sage: E = EllipticCurve('11a1'); P = E.kolyvagin_point(-7)
sage: P.mod(3,70)  # long time (4s on sage.math, 2013)
(1 : 2 : 1)
sage: P.mod(5,70)
(1 : 4 : 1)
sage: P.mod(7,70)
Traceback (most recent call last):
...
ValueError: p must be coprime to conductors and discriminant
sage: P.mod(11,70)
Traceback (most recent call last):
...
ValueError: p must be coprime to conductors and discriminant
sage: P.mod(13,70)
(3 : 4 : 1)

REFERENCES:

[SteinToward]Stein, “Toward a Generalization of the Gross-Zagier Conjecture”, Int Math Res Notices (2011), doi:10.1093/imrn/rnq075
numerical_approx(prec=53)

Return a numerical approximation to this Kolyvagin point using prec bits of working precision.

INPUT:

  • prec – precision in bits (default: 53)

EXAMPLES:

sage: P = EllipticCurve('37a1').kolyvagin_point(-7); P
Kolyvagin point of discriminant -7 on elliptic curve of conductor 37
sage: P.numerical_approx() # approx. (0 : 0 : 1)
(...e-16 - ...e-16*I : ...e-16 + ...e-16*I : 1.00000000000000)
sage: P.numerical_approx(100)[0].abs() < 2.0^-99
True

sage: P = EllipticCurve('389a1').kolyvagin_point(-7, 5); P
Kolyvagin point of discriminant -7 and conductor 5 on elliptic curve of conductor 389

Numerical approximation is only implemented for points of conductor 1:

sage: P.numerical_approx()
Traceback (most recent call last):
...
NotImplementedError
plot(prec=53, *args, **kwds)

Plot a Kolyvagin point \(P_1\) if it is defined over the rational numbers.

EXAMPLES:

sage: E = EllipticCurve('37a'); P = E.heegner_point(-11).kolyvagin_point()
sage: P.plot(prec=30, pointsize=50, rgbcolor='red') + E.plot()
point_exact(prec=53)

INPUT:

  • prec – precision in bits (default: 53)

EXAMPLES:

A rank 1 curve:

sage: E = EllipticCurve('37a1'); P = E.kolyvagin_point(-67)
sage: P.point_exact()
(6 : -15 : 1)
sage: P.point_exact(40)
(6 : -15 : 1)
sage: P.point_exact(20)
Traceback (most recent call last):
...
RuntimeError: insufficient precision to find exact point

A rank 0 curve:

sage: E = EllipticCurve('11a1'); P = E.kolyvagin_point(-7)
sage: P.point_exact()
(-1/2*sqrt_minus_7 + 1/2 : -2*sqrt_minus_7 - 2 : 1)

A rank 2 curve:

sage: E = EllipticCurve('389a1'); P = E.kolyvagin_point(-7)
sage: P.point_exact()
(0 : 1 : 0)
satisfies_kolyvagin_hypothesis(n=None)

Return True if this Kolyvagin point satisfies the Heegner hypothesis for \(n\), so that it defines a Galois equivariant element of \(E(K_c)/n E(K_c)\).

EXAMPLES:

sage: y = EllipticCurve('389a').heegner_point(-7,5); P = y.kolyvagin_point()
sage: P.kolyvagin_cohomology_class(3)
Kolyvagin cohomology class c(5) in H^1(K,E[3])
sage: P.satisfies_kolyvagin_hypothesis(3)
True
sage: P.satisfies_kolyvagin_hypothesis(5)
False
sage: P.satisfies_kolyvagin_hypothesis(7)
False
sage: P.satisfies_kolyvagin_hypothesis(11)
False
trace_to_real_numerical(prec=53)

Return the trace of this Kolyvagin point down to the real numbers, computed numerically using prec bits of working precision.

EXAMPLES:

sage: E = EllipticCurve('37a1'); P = E.kolyvagin_point(-67)
sage: PP = P.numerical_approx(); PP
(6.00000000000000 ... : -15.0000000000000 ... : 1.00000000000000)
sage: [c.real() for c in PP]
[6.00000000000000, -15.0000000000000, 1.00000000000000]
sage: all([c.imag().abs() < 1e-14 for c in PP])
True
sage: P.trace_to_real_numerical()
(1.61355529131986 : -2.18446840788880 : 1.00000000000000)
sage: P.trace_to_real_numerical(prec=80)
(1.6135552913198573127230 : -2.1844684078888023289187 : 1.0000000000000000000000)
class sage.schemes.elliptic_curves.heegner.RingClassField(D, c, check=True)

Bases: sage.structure.sage_object.SageObject

A Ring class field of a quadratic imaginary field of given conductor.

Note

This is a ring class field, not a ray class field. In general, the ring class field of given conductor is a subfield of the ray class field of the same conductor.

EXAMPLES:

sage: heegner_point(37,-7).ring_class_field()
Hilbert class field of QQ[sqrt(-7)]
sage: heegner_point(37,-7,5).ring_class_field()
Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: heegner_point(37,-7,55).ring_class_field()
Ring class field extension of QQ[sqrt(-7)] of conductor 55

TESTS:

sage: K_c = heegner_point(37,-7).ring_class_field()
sage: type(K_c)
<class 'sage.schemes.elliptic_curves.heegner.RingClassField'>
sage: loads(dumps(K_c)) == K_c
True
absolute_degree()

Return the absolute degree of this field over \(\QQ\).

EXAMPLES:

sage: E = EllipticCurve('389a'); K = E.heegner_point(-7,5).ring_class_field()
sage: K.absolute_degree()
12
sage: K.degree_over_K()
6
conductor()

Return the conductor of this ring class field.

EXAMPLES:

sage: E = EllipticCurve('389a'); K5 = E.heegner_point(-7,5).ring_class_field()
sage: K5.conductor()
5
degree_over_H()

Return the degree of this field over the Hilbert class field \(H\) of \(K\).

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: E.heegner_point(-59).ring_class_field().degree_over_H()
1
sage: E.heegner_point(-59).ring_class_field().degree_over_K()
3
sage: QuadraticField(-59,'a').class_number()
3

Some examples in which prime dividing c is inert:

sage: heegner_point(37,-7,3).ring_class_field().degree_over_H()
4
sage: heegner_point(37,-7,3^2).ring_class_field().degree_over_H()
12
sage: heegner_point(37,-7,3^3).ring_class_field().degree_over_H()
36

The prime dividing c is split. For example, in the first case \(O_K/cO_K\) is isomorphic to a direct sum of two copies of GF(2), so the units are trivial:

sage: heegner_point(37,-7,2).ring_class_field().degree_over_H()
1
sage: heegner_point(37,-7,4).ring_class_field().degree_over_H()
2
sage: heegner_point(37,-7,8).ring_class_field().degree_over_H()
4

Now c is ramified:

sage: heegner_point(37,-7,7).ring_class_field().degree_over_H()
7
sage: heegner_point(37,-7,7^2).ring_class_field().degree_over_H()
49

Check that trac ticket #15218 is solved:

sage: E = EllipticCurve("19a");
sage: s = E.heegner_point(-3,2).ring_class_field().galois_group().complex_conjugation()
sage: H = s.domain(); H.absolute_degree()
2
degree_over_K()

Return the relative degree of this ring class field over the quadratic imaginary field \(K\).

EXAMPLES:

sage: E = EllipticCurve('389a'); P = E.heegner_point(-7,5)
sage: K5 = P.ring_class_field(); K5
Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: K5.degree_over_K()
6
sage: type(K5.degree_over_K())
<type 'sage.rings.integer.Integer'>

sage: E = EllipticCurve('389a'); E.heegner_point(-20).ring_class_field().degree_over_K()
2
sage: E.heegner_point(-20,3).ring_class_field().degree_over_K()
4
sage: kronecker(-20,11)
-1
sage: E.heegner_point(-20,11).ring_class_field().degree_over_K()
24
degree_over_Q()

Return the absolute degree of this field over \(\QQ\).

EXAMPLES:

sage: E = EllipticCurve('389a'); K = E.heegner_point(-7,5).ring_class_field()
sage: K.absolute_degree()
12
sage: K.degree_over_K()
6
discriminant_of_K()

Return the discriminant of the quadratic imaginary field \(K\) contained in self.

EXAMPLES:

sage: E = EllipticCurve('389a'); K5 = E.heegner_point(-7,5).ring_class_field()
sage: K5.discriminant_of_K()
-7
galois_group(base=Rational Field)

Return the Galois group of self over base.

INPUT:

  • base – (default: \(\QQ\)) a subfield of self or \(\QQ\)

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: A = E.heegner_point(-7,5).ring_class_field()
sage: A.galois_group()
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: B = E.heegner_point(-7).ring_class_field()
sage: C = E.heegner_point(-7,15).ring_class_field()
sage: A.galois_group()
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: A.galois_group(B)
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5 over Hilbert class field of QQ[sqrt(-7)]
sage: A.galois_group().cardinality()
12
sage: A.galois_group(B).cardinality()
6
sage: C.galois_group(A)
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 15 over Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: C.galois_group(A).cardinality()
4
is_subfield(M)

Return True if this ring class field is a subfield of the ring class field \(M\). If \(M\) is not a ring class field, then a TypeError is raised.

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: A = E.heegner_point(-7,5).ring_class_field()
sage: B = E.heegner_point(-7).ring_class_field()
sage: C = E.heegner_point(-20).ring_class_field()
sage: D = E.heegner_point(-7,15).ring_class_field()
sage: B.is_subfield(A)
True
sage: B.is_subfield(B)
True
sage: B.is_subfield(D)
True
sage: B.is_subfield(C)
False
sage: A.is_subfield(B)
False
sage: A.is_subfield(D)
True
quadratic_field()

Return the quadratic imaginary field \(K = \QQ(\sqrt{D})\).

EXAMPLES:

sage: E = EllipticCurve('389a'); K = E.heegner_point(-7,5).ring_class_field()
sage: K.quadratic_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
ramified_primes()

Return the primes of \(\ZZ\) that ramify in this ring class field.

EXAMPLES:

sage: E = EllipticCurve('389a'); K55 = E.heegner_point(-7,55).ring_class_field()
sage: K55.ramified_primes()
[5, 7, 11]
sage: E.heegner_point(-7).ring_class_field().ramified_primes()
[7]
sage.schemes.elliptic_curves.heegner.class_number(D)

Return the class number of the quadratic field with fundamental discriminant \(D\).

INPUT:

  • \(D\) – integer

EXAMPLES:

sage: sage.schemes.elliptic_curves.heegner.class_number(-20)
2
sage: sage.schemes.elliptic_curves.heegner.class_number(-23)
3
sage: sage.schemes.elliptic_curves.heegner.class_number(-163)
1

A ValueError is raised when \(D\) is not a fundamental discriminant:

sage: sage.schemes.elliptic_curves.heegner.class_number(-5)
Traceback (most recent call last):
...
ValueError: D (=-5) must be a fundamental discriminant
sage.schemes.elliptic_curves.heegner.ell_heegner_discriminants(self, bound)

Return the list of self’s Heegner discriminants between -1 and -bound.

INPUT:

  • bound (int) - upper bound for -discriminant

OUTPUT: The list of Heegner discriminants between -1 and -bound for the given elliptic curve.

EXAMPLES:

sage: E=EllipticCurve('11a')
sage: E.heegner_discriminants(30)                     # indirect doctest
[-7, -8, -19, -24]
sage.schemes.elliptic_curves.heegner.ell_heegner_discriminants_list(self, n)

Return the list of self’s first \(n\) Heegner discriminants smaller than -5.

INPUT:

  • n (int) - the number of discriminants to compute

OUTPUT: The list of the first n Heegner discriminants smaller than -5 for the given elliptic curve.

EXAMPLE:

sage: E=EllipticCurve('11a')
sage: E.heegner_discriminants_list(4)                     # indirect doctest
[-7, -8, -19, -24]
sage.schemes.elliptic_curves.heegner.ell_heegner_point(self, D, c=1, f=None, check=True)

Returns the Heegner point on this curve associated to the quadratic imaginary field \(K=\QQ(\sqrt{D})\).

If the optional parameter \(c\) is given, returns the higher Heegner point associated to the order of conductor \(c\).

INPUT:

- `D`        -- a Heegner discriminant

- `c`        -- (default: 1) conductor, must be coprime to `DN`

- `f`        -- binary quadratic form or 3-tuple `(A,B,C)` of coefficients
                of `AX^2 + BXY + CY^2`

- ``check``  -- bool (default: ``True``)

OUTPUT:

The Heegner point `y_c`.

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: E.heegner_discriminants_list(10)
[-7, -11, -40, -47, -67, -71, -83, -84, -95, -104]
sage: P = E.heegner_point(-7); P                          # indirect doctest
Heegner point of discriminant -7 on elliptic curve of conductor 37
sage: P.point_exact()
(0 : 0 : 1)
sage: P.curve()
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: P = E.heegner_point(-40).point_exact(); P
(a : -a + 1 : 1)
sage: P = E.heegner_point(-47).point_exact(); P
(a : a^4 + a - 1 : 1)
sage: P[0].parent()
Number Field in a with defining polynomial x^5 - x^4 + x^3 + x^2 - 2*x + 1

Working out the details manually:

sage: P = E.heegner_point(-47).numerical_approx(prec=200)
sage: f = algdep(P[0], 5); f
x^5 - x^4 + x^3 + x^2 - 2*x + 1
sage: f.discriminant().factor()
47^2

The Heegner hypothesis is checked:

sage: E = EllipticCurve('389a'); P = E.heegner_point(-5,7);
Traceback (most recent call last):
...
ValueError: N (=389) and D (=-5) must satisfy the Heegner hypothesis

We can specify the quadratic form:

sage: P = EllipticCurve('389a').heegner_point(-7, 5, (778,925,275)); P
Heegner point of discriminant -7 and conductor 5 on elliptic curve of conductor 389
sage: P.quadratic_form()
778*x^2 + 925*x*y + 275*y^2
sage.schemes.elliptic_curves.heegner.heegner_index(self, D, min_p=2, prec=5, descent_second_limit=12, verbose_mwrank=False, check_rank=True)

Return an interval that contains the index of the Heegner point \(y_K\) in the group of \(K\)-rational points modulo torsion on this elliptic curve, computed using the Gross-Zagier formula and/or a point search, or possibly half the index if the rank is greater than one.

If the curve has rank > 1, then the returned index is infinity.

Note

If min_p is bigger than 2 then the index can be off by any prime less than min_p. This function returns the index divided by \(2\) exactly when the rank of \(E(K)\) is greater than 1 and \(E(\QQ)_{/tor} \oplus E^D(\QQ)_{/tor}\) has index \(2\) in \(E(K)_{/tor}\), where the second factor undergoes a twist.

INPUT:

  • D (int) - Heegner discriminant
  • min_p (int) - (default: 2) only rule out primes = min_p dividing the index.
  • verbose_mwrank (bool) - (default: False); print lots of mwrank search status information when computing regulator
  • prec (int) - (default: 5), use prec*sqrt(N) + 20 terms of L-series in computations, where N is the conductor.
  • descent_second_limit - (default: 12)- used in 2-descent when computing regulator of the twist
  • check_rank - whether to check if the rank is at least 2 by computing the Mordell-Weil rank directly.

OUTPUT: an interval that contains the index, or half the index

EXAMPLES:

sage: E = EllipticCurve('11a')
sage: E.heegner_discriminants(50)
[-7, -8, -19, -24, -35, -39, -40, -43]
sage: E.heegner_index(-7)
1.00000?
sage: E = EllipticCurve('37b')
sage: E.heegner_discriminants(100)
[-3, -4, -7, -11, -40, -47, -67, -71, -83, -84, -95]
sage: E.heegner_index(-95)          # long time (1 second)
2.00000?

This tests doing direct computation of the Mordell-Weil group.

sage: EllipticCurve('675b').heegner_index(-11)
3.0000?

Currently discriminants -3 and -4 are not supported:

sage: E.heegner_index(-3)
Traceback (most recent call last):
...
ArithmeticError: Discriminant (=-3) must not be -3 or -4.

The curve 681b returns the true index, which is \(3\):

sage: E = EllipticCurve('681b')
sage: I = E.heegner_index(-8); I
3.0000?

In fact, whenever the returned index has a denominator of \(2\), the true index is got by multiplying the returned index by \(2\). Unfortunately, this is not an if and only if condition, i.e., sometimes the index must be multiplied by \(2\) even though the denominator is not \(2\).

This example demonstrates the descent_second_limit option, which can be used to fine tune the 2-descent used to compute the regulator of the twist:

sage: E = EllipticCurve([0, 0, 1, -34874, -2506691])
sage: E.heegner_index(-8)
Traceback (most recent call last):
...
RuntimeError: ...

However when we search higher, we find the points we need:

sage: E.heegner_index(-8, descent_second_limit=16, check_rank=False)
1.00000?

Two higher rank examples (of ranks 2 and 3):

sage: E = EllipticCurve('389a')
sage: E.heegner_index(-7)
+Infinity
sage: E = EllipticCurve('5077a')
sage: E.heegner_index(-7)
+Infinity
sage: E.heegner_index(-7, check_rank=False)
0.001?
sage: E.heegner_index(-7, check_rank=False).lower() == 0
True
sage.schemes.elliptic_curves.heegner.heegner_index_bound(self, D=0, prec=5, max_height=None)

Assume self has rank 0.

Return a list \(v\) of primes such that if an odd prime \(p\) divides the index of the Heegner point in the group of rational points modulo torsion, then \(p\) is in \(v\).

If 0 is in the interval of the height of the Heegner point computed to the given prec, then this function returns \(v = 0\). This does not mean that the Heegner point is torsion, just that it is very likely torsion.

If we obtain no information from a search up to max_height, e.g., if the Siksek et al. bound is bigger than max_height, then we return \(v = -1\).

INPUT:

  • D (int) - (default: 0) Heegner discriminant; if 0, use the first discriminant -4 that satisfies the Heegner hypothesis
  • verbose (bool) - (default: True)
  • prec (int) - (default: 5), use \(prec \cdot \sqrt(N) + 20\) terms of \(L\)-series in computations, where \(N\) is the conductor.
  • max_height (float) - should be = 21; bound on logarithmic naive height used in point searches. Make smaller to make this function faster, at the expense of possibly obtaining a worse answer. A good range is between 13 and 21.

OUTPUT:

  • v - list or int (bad primes or 0 or -1)
  • D - the discriminant that was used (this is useful if \(D\) was automatically selected).
  • exact - either False, or the exact Heegner index (up to factors of 2)

EXAMPLES:

sage: E = EllipticCurve('11a1')
sage: E.heegner_index_bound()
([2], -7, 2)
sage.schemes.elliptic_curves.heegner.heegner_point(N, D=None, c=1)

Return a specific Heegner point of level \(N\) with given discriminant and conductor. If \(D\) is not specified, then the first valid Heegner discriminant is used. If \(c\) is not given, then \(c=1\) is used.

INPUT:

  • \(N\) – level (positive integer)
  • \(D\) – discriminant (optional: default first valid \(D\))
  • \(c\) – conductor (positive integer, optional, default: 1)

EXAMPLES:

sage: heegner_point(389)
Heegner point 1/778*sqrt(-7) - 185/778 of discriminant -7 on X_0(389)
sage: heegner_point(389,-7)
Heegner point 1/778*sqrt(-7) - 185/778 of discriminant -7 on X_0(389)
sage: heegner_point(389,-7,5)
Heegner point 5/778*sqrt(-7) - 147/778 of discriminant -7 and conductor 5 on X_0(389)
sage: heegner_point(389,-20)
Heegner point 1/778*sqrt(-20) - 165/389 of discriminant -20 on X_0(389)
sage.schemes.elliptic_curves.heegner.heegner_point_height(self, D, prec=2, check_rank=True)

Use the Gross-Zagier formula to compute the Neron-Tate canonical height over \(K\) of the Heegner point corresponding to \(D\), as an interval (it is computed to some precision using \(L\)-functions).

If the curve has rank at least 2, then the returned height is the exact Sage integer 0.

INPUT:

  • D (int) - fundamental discriminant (=/= -3, -4)

  • prec (int) - (default: 2), use \(prec \cdot \sqrt(N) + 20\)

    terms of \(L\)-series in computations, where \(N\) is the conductor.

  • check_rank - whether to check if the rank is at least 2 by computing the Mordell-Weil rank directly.

OUTPUT: Interval that contains the height of the Heegner point.

EXAMPLE:

sage: E = EllipticCurve('11a')
sage: E.heegner_point_height(-7)
0.22227?

Some higher rank examples:

sage: E = EllipticCurve('389a')
sage: E.heegner_point_height(-7)
0
sage: E = EllipticCurve('5077a')
sage: E.heegner_point_height(-7)
0
sage: E.heegner_point_height(-7,check_rank=False)
0.0000?
sage.schemes.elliptic_curves.heegner.heegner_points(N, D=None, c=None)

Return all Heegner points of given level \(N\). Can also restrict to Heegner points with specified discriminant \(D\) and optionally conductor \(c\).

INPUT:

  • \(N\) – level (positive integer)
  • \(D\) – discriminant (negative integer)
  • \(c\) – conductor (positive integer)

EXAMPLES:

sage: heegner_points(389,-7)
Set of all Heegner points on X_0(389) associated to QQ[sqrt(-7)]
sage: heegner_points(389,-7,1)
All Heegner points of conductor 1 on X_0(389) associated to QQ[sqrt(-7)]
sage: heegner_points(389,-7,5)
All Heegner points of conductor 5 on X_0(389) associated to QQ[sqrt(-7)]
sage.schemes.elliptic_curves.heegner.heegner_sha_an(self, D, prec=53)

Return the conjectural (analytic) order of Sha for E over the field \(K=\QQ(\sqrt{D})\).

INPUT:

  • \(D\) – negative integer; the Heegner discriminant
  • prec – integer (default: 53); bits of precision to compute analytic order of Sha

OUTPUT:

(floating point number) an approximation to the conjectural order of Sha.

Note

Often you’ll want to do proof.elliptic_curve(False) when using this function, since often the twisted elliptic curves that come up have enormous conductor, and Sha is nontrivial, which makes provably finding the Mordell-Weil group using 2-descent difficult.

EXAMPLES:

An example where E has conductor 11:

sage: E = EllipticCurve('11a')
sage: E.heegner_sha_an(-7)                                  # long time
1.00000000000000

The cache works:

sage: E.heegner_sha_an(-7) is E.heegner_sha_an(-7)          # long time
True

Lower precision:

sage: E.heegner_sha_an(-7,10)                               # long time
1.0

Checking that the cache works for any precision:

sage: E.heegner_sha_an(-7,10) is E.heegner_sha_an(-7,10)    # long time
True

Next we consider a rank 1 curve with nontrivial Sha over the quadratic imaginary field \(K\); however, there is no Sha for \(E\) over \(\QQ\) or for the quadratic twist of \(E\):

sage: E = EllipticCurve('37a')
sage: E.heegner_sha_an(-40)                                 # long time
4.00000000000000
sage: E.quadratic_twist(-40).sha().an()                     # long time
1
sage: E.sha().an()                                          # long time
1

A rank 2 curve:

sage: E = EllipticCurve('389a')                             # long time
sage: E.heegner_sha_an(-7)                                  # long time
1.00000000000000

If we remove the hypothesis that \(E(K)\) has rank 1 in Conjecture 2.3 in [Gross-Zagier, 1986, page 311], then that conjecture is false, as the following example shows:

sage: E = EllipticCurve('65a')                              # long time
sage: E.heegner_sha_an(-56)                                 # long time
1.00000000000000
sage: E.torsion_order()                                     # long time
2
sage: E.tamagawa_product()                                  # long time
1
sage: E.quadratic_twist(-56).rank()                         # long time
2
sage.schemes.elliptic_curves.heegner.is_inert(D, p)

Return True if p is an inert prime in the field \(\QQ(\sqrt{D})\).

INPUT:

  • \(D\) – fundamental discriminant
  • \(p\) – prime integer

EXAMPLES:

sage: sage.schemes.elliptic_curves.heegner.is_inert(-7,3)
True
sage: sage.schemes.elliptic_curves.heegner.is_inert(-7,7)
False
sage: sage.schemes.elliptic_curves.heegner.is_inert(-7,11)
False
sage.schemes.elliptic_curves.heegner.is_kolyvagin_conductor(N, E, D, r, n, c)

Return True if \(c\) is a Kolyvagin conductor for level \(N\), discriminant \(D\), mod \(n\), etc., i.e., \(c\) is divisible by exactly \(r\) prime factors, is coprime to \(ND\), each prime dividing \(c\) is inert, and if \(E\) is not None then \(n | \gcd(p+1, a_p(E))\) for each prime \(p\) dividing \(c\).

INPUT:

  • \(N\) – level (positive integer)
  • \(E\) – elliptic curve or None
  • \(D\) – negative fundamental discriminant
  • \(r\) – number of prime factors (nonnegative integer) or None
  • \(n\) – torsion order (i.e., do we get class in \((E(K_c)/n E(K_c))^{Gal(K_c/K)}\)?)
  • \(c\) – conductor (positive integer)

EXAMPLES:

sage: from sage.schemes.elliptic_curves.heegner import is_kolyvagin_conductor
sage: is_kolyvagin_conductor(389,None,-7,1,None,5)
True
sage: is_kolyvagin_conductor(389,None,-7,1,None,7)
False
sage: is_kolyvagin_conductor(389,None,-7,1,None,11)
False
sage: is_kolyvagin_conductor(389,EllipticCurve('389a'),-7,1,3,5)
True
sage: is_kolyvagin_conductor(389,EllipticCurve('389a'),-7,1,11,5)
False
sage.schemes.elliptic_curves.heegner.is_ramified(D, p)

Return True if p is a ramified prime in the field \(\QQ(\sqrt{D})\).

INPUT:

  • \(D\) – fundamental discriminant
  • \(p\) – prime integer

EXAMPLES:

sage: sage.schemes.elliptic_curves.heegner.is_ramified(-7,2)
False
sage: sage.schemes.elliptic_curves.heegner.is_ramified(-7,7)
True
sage: sage.schemes.elliptic_curves.heegner.is_ramified(-1,2)
True
sage.schemes.elliptic_curves.heegner.is_split(D, p)

Return True if p is a split prime in the field \(\QQ(\sqrt{D})\).

INPUT:

  • \(D\) – fundamental discriminant
  • \(p\) – prime integer

EXAMPLES:

sage: sage.schemes.elliptic_curves.heegner.is_split(-7,3)
False
sage: sage.schemes.elliptic_curves.heegner.is_split(-7,7)
False
sage: sage.schemes.elliptic_curves.heegner.is_split(-7,11)
True
sage.schemes.elliptic_curves.heegner.kolyvagin_point(self, D, c=1, check=True)

Returns the Kolyvagin point on this curve associated to the quadratic imaginary field \(K=\QQ(\sqrt{D})\) and conductor \(c\).

INPUT:

  • \(D\) – a Heegner discriminant
  • \(c\) – (default: 1) conductor, must be coprime to \(DN\)
  • check – bool (default: True)

OUTPUT:

The Kolyvagin point \(P\) of conductor \(c\).

EXAMPLES:

sage: E = EllipticCurve('37a1')
sage: P = E.kolyvagin_point(-67); P
Kolyvagin point of discriminant -67 on elliptic curve of conductor 37
sage: P.numerical_approx() # imaginary parts approx. 0
(6.00000000000000 ... : -15.0000000000000 ... : 1.00000000000000)
sage: P.index()
6
sage: g = E((0,-1,1)) # a generator
sage: E.regulator() == E.regulator_of_points([g])
True
sage: 6*g
(6 : -15 : 1)
sage.schemes.elliptic_curves.heegner.kolyvagin_reduction_data(E, q, first_only=True)

Given an elliptic curve of positive rank and a prime \(q\), this function returns data about how to use Kolyvagin’s \(q\)-torsion Heegner point Euler system to do computations with this curve. See the precise description of the output below.

INPUT:

  • \(E\) – elliptic curve over \(\QQ\) of rank 1 or 2

  • \(q\) – an odd prime that does not divide the order of the

    rational torsion subgroup of \(E\)

  • first_only – bool (default: True) whether two only return

    the first prime that one can work modulo to get data about the Euler system

OUTPUT in the rank 1 case or when the default flag first_only=True:

  • \(\ell\) – first good odd prime satisfying the Kolyvagin

    condition that \(q\) divides gcd(a_{ell},ell+1)` and the reduction map is surjective to \(E(\GF{\ell}) / q E(\GF{\ell})\)

  • \(D\) – discriminant of the first quadratic imaginary field

    \(K\) that satisfies the Heegner hypothesis for \(E\) such that both \(\ell\) is inert in \(K\), and the twist \(E^D\) has analytic rank \(\leq 1\)

  • \(h_D\) – the class number of \(K\)

  • the dimension of the Brandt module \(B(\ell,N)\), where \(N\) is the conductor of \(E\)

OUTPUT in the rank 2 case:

  • \(\ell_1\) – first prime (as above in the rank 1 case) where reduction map is surjective

  • \(\ell_2\) – second prime (as above) where reduction map is surjective

  • \(D\) – discriminant of the first quadratic imaginary field

    \(K\) that satisfies the Heegner hypothesis for \(E\) such that both \(\ell_1\) and \(\ell_2\) are simultaneously inert in \(K\), and the twist \(E^D\) has analytic rank \(\leq 1\)

  • \(h_D\) – the class number of \(K\)

  • the dimension of the Brandt module \(B(\ell_1,N)\), where \(N\) is the conductor of \(E\)

  • the dimension of the Brandt module \(B(\ell_2,N)\)

EXAMPLES:

Import this function:

sage: from sage.schemes.elliptic_curves.heegner import kolyvagin_reduction_data

A rank 1 example:

sage: kolyvagin_reduction_data(EllipticCurve('37a1'),3)
(17, -7, 1, 52)

A rank 3 example:

sage: kolyvagin_reduction_data(EllipticCurve('5077a1'),3)
(11, -47, 5, 4234)
sage: H = heegner_points(5077, -47)
sage: [c for c in H.kolyvagin_conductors(2,10,EllipticCurve('5077a1'),3) if c%11]
[667, 943, 1189, 2461]
sage: factor(667)
23 * 29

A rank 4 example (the first Kolyvagin class that we could try to compute would be \(P_{23\cdot 29\cdot 41}\), and would require working in a space of dimension 293060 (so prohibitive at present):

sage: E = elliptic_curves.rank(4)[0]
sage: kolyvagin_reduction_data(E,3)              # long time
(11, -71, 7, 293060)
sage: H = heegner_points(293060, -71)
sage: H.kolyvagin_conductors(1,4,E,3)
[11, 17, 23, 41]

The first rank 2 example:

sage: kolyvagin_reduction_data(EllipticCurve('389a'),3)
(5, -7, 1, 130)
sage: kolyvagin_reduction_data(EllipticCurve('389a'),3, first_only=False)
(5, 17, -7, 1, 130, 520)

A large \(q = 7\):

sage: kolyvagin_reduction_data(EllipticCurve('1143c1'),7, first_only=False)
(13, 83, -59, 3, 1536, 10496)

Additive reduction:

sage: kolyvagin_reduction_data(EllipticCurve('2350g1'),5, first_only=False)
(19, 239, -311, 19, 6480, 85680)
sage.schemes.elliptic_curves.heegner.make_monic(f)

make_monic returns a monic integral polynomial \(g\) and an integer \(d\) such that if \(\alpha\) is a root of \(g\) then a root of \(f\) is \(\alpha/d\).

INPUT:

  • f – polynomial over the rational numbers

EXAMPLES:

sage: R.<x> = QQ[]
sage: sage.schemes.elliptic_curves.heegner.make_monic(3*x^3 + 14*x^2 - 7*x + 5)
(x^3 + 14*x^2 - 21*x + 45, 3)

In this example we verify that make_monic does what we claim it does:

sage: K.<a> = NumberField(x^3 + 17*x - 3)
sage: f = (a/7+2/3).minpoly(); f
x^3 - 2*x^2 + 247/147*x - 4967/9261
sage: g, d = sage.schemes.elliptic_curves.heegner.make_monic(f)
sage: g
x^3 - 18522*x^2 + 144110421*x - 426000323007
sage: d
9261
sage: K.<b> = NumberField(g)
sage: (b/d).minpoly()
x^3 - 2*x^2 + 247/147*x - 4967/9261
sage.schemes.elliptic_curves.heegner.nearby_rational_poly(f, **kwds)

Return a polynomial whose coefficients are rational numbers close to the coefficients of \(f\).

INPUT:

  • \(f\) – polynomial with real floating point entries
  • **kwds – passed on to nearby_rational method

EXAMPLES:

sage: R.<x> = RR[]
sage: sage.schemes.elliptic_curves.heegner.nearby_rational_poly(2.1*x^2 + 3.5*x - 1.2, max_error=10e-16)
21/10*X^2 + 7/2*X - 6/5
sage: sage.schemes.elliptic_curves.heegner.nearby_rational_poly(2.1*x^2 + 3.5*x - 1.2, max_error=10e-17)
4728779608739021/2251799813685248*X^2 + 7/2*X - 5404319552844595/4503599627370496
sage: RR(4728779608739021/2251799813685248  - 21/10)
8.88178419700125e-17
sage.schemes.elliptic_curves.heegner.quadratic_order(D, c, names='a')

Return order of conductor \(c\) in quadratic field with fundamental discriminant \(D\).

INPUT:

  • \(D\) – fundamental discriminant
  • \(c\) – conductor
  • names – string (default: ‘a’)

OUTPUT:

  • order \(R\) of conductor \(c\) in an imaginary quadratic field
  • the element \(c\sqrt{D}\) as an element of \(R\)

The generator for the field is named ‘a’ by default.

EXAMPLES:

sage: sage.schemes.elliptic_curves.heegner.quadratic_order(-7,3)
(Order in Number Field in a with defining polynomial x^2 + 7, 3*a)
sage: sage.schemes.elliptic_curves.heegner.quadratic_order(-7,3,'alpha')
(Order in Number Field in alpha with defining polynomial x^2 + 7, 3*alpha)
sage.schemes.elliptic_curves.heegner.satisfies_heegner_hypothesis(self, D)

Returns True precisely when \(D\) is a fundamental discriminant that satisfies the Heegner hypothesis for this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve('11a1')
sage: E.satisfies_heegner_hypothesis(-7)
True
sage: E.satisfies_heegner_hypothesis(-11)
False
sage.schemes.elliptic_curves.heegner.satisfies_weak_heegner_hypothesis(N, D)

Check that \(D\) satisfies the weak Heegner hypothesis relative to \(N\). This is all that is needed to define Heegner points.

The condition is that \(D<0\) is a fundamental discriminant and that each unramified prime dividing \(N\) splits in \(K=\QQ(\sqrt{D})\) and each ramified prime exactly divides \(N\). We also do not require that \(D<-4\).

INPUT:

  • \(N\) – positive integer
  • \(D\) – negative integer

EXAMPLES:

sage: s = sage.schemes.elliptic_curves.heegner.satisfies_weak_heegner_hypothesis
sage: s(37,-7)
True
sage: s(37,-37)
False
sage: s(37,-37*4)
True
sage: s(100,-4)
False
sage: [D for D in [-1,-2,..,-40] if s(37,D)]
[-3, -4, -7, -11, -40]
sage: [D for D in [-1,-2,..,-100] if s(37,D)]
[-3, -4, -7, -11, -40, -47, -67, -71, -83, -84, -95]
sage: EllipticCurve('37a').heegner_discriminants_list(10)
[-7, -11, -40, -47, -67, -71, -83, -84, -95, -104]
sage.schemes.elliptic_curves.heegner.simplest_rational_poly(f, prec)

Return a polynomial whose coefficients are as simple as possible rationals that are also close to the coefficients of f.

INPUT:

  • \(f\) – polynomial with real floating point entries
  • prec – positive integer

EXAMPLES:

sage: R.<x> = RR[]
sage: sage.schemes.elliptic_curves.heegner.simplest_rational_poly(2.1*x^2 + 3.5*x - 1.2, 53)
21/10*X^2 + 7/2*X - 6/5

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