We define our elliptic curve, which has rank 1:
{{{
E = EllipticCurve('37a')
show(E)
show(E.plot(hue=0.8, thickness=3))
///
y^2 + y = x^3 - x
}}}
We compute a list of the first 20 fundamental discriminants that
satisfy the Heegner hypothesis.
{{{
discs = E.heegner_discriminants_list(20); discs
///
[-7, -11, -40, -47, -67, -71, -83, -84, -95, -104, -107, -115, -120, -123, -127, -132, -136, -139, -151, -152]
}}}
Table
Here is a nice table that gives each discriminant, the class number,
the index of the Heegner point in $E(K)$, and the actual Heegner point
(up to sign). The Heegner point is computed by cheating -- we compute
the height of the Heegner point using the Gross-Zagier formula, and
simply take the corresponding multiple of a known generator of $E$.
{{{
P = E.gens()[0]
print ''
print '| Discriminant | Class Number | Index | Point |
'
for D in discs:
ind = E.heegner_index(D)
if ind > 0:
ind = ind.sqrt().simplest_rational()
y = ZZ(ind)*P
else:
ind = infinity
y = E(0)
hK = QuadraticField(D,'sqrtD').class_number()
print '| %s | %s | %s | %s |
'%(D,hK,ind,y)
print '
'
///
| Discriminant | Class Number | Index | Point |
| -7 | 1 | 1 | (0 : 0 : 1) |
| -11 | 1 | 1 | (0 : 0 : 1) |
| -40 | 2 | 2 | (1 : 0 : 1) |
| -47 | 5 | 1 | (0 : 0 : 1) |
| -67 | 1 | 6 | (6 : 14 : 1) |
| -71 | 7 | 1 | (0 : 0 : 1) |
| -83 | 3 | 1 | (0 : 0 : 1) |
| -84 | 4 | 1 | (0 : 0 : 1) |
| -95 | 8 | +Infinity | (0 : 1 : 0) |
| -104 | 6 | +Infinity | (0 : 1 : 0) |
| -107 | 3 | +Infinity | (0 : 1 : 0) |
| -115 | 2 | 6 | (6 : 14 : 1) |
| -120 | 4 | 2 | (1 : 0 : 1) |
| -123 | 2 | 3 | (-1 : -1 : 1) |
| -127 | 5 | 1 | (0 : 0 : 1) |
| -132 | 4 | 3 | (-1 : -1 : 1) |
| -136 | 4 | 4 | (2 : -3 : 1) |
| -139 | 3 | +Infinity | (0 : 1 : 0) |
| -151 | 7 | 2 | (1 : 0 : 1) |
| -152 | 6 | 2 | (1 : 0 : 1) |
}}}
Now we directly compute the Heegner point for discriminant -7
{{{
K. = QuadraticField(-7); K
///
Number Field in a with defining polynomial x^2 + 7
}}}
{{{
D = K.discriminant(); D
///
-7
}}}
{{{
K.class_group()
///
Trivial Abelian Group
}}}
{{{
N = 37
}}}
We compute the square roots of D modulo $4N$.
{{{
srs = Mod(D, 4*N).sqrt(all=True); srs
///
[17, 57, 91, 131]
}}}
Then make a list of their reductions modulo $2N$:
{{{
S = list(set([Mod(a, 2*N) for a in srs])); S
///
[17, 57]
}}}
For fun we list all representative primitive binary quadratic
forms of discriminant -7 -- we will not need this later.
{{{
C = BinaryQF_reduced_representatives(-7); C
///
[x^2 + xy + 2y^2 ]
}}}
The following is useful for making the computations more efficient; we
will not use it yet.
{{{
def minimal_form(f, N):
"""
Given a positive definite form f = (A,B,C) of discriminant D < 0
with N | A, this function returns a form f' = (A',B',C') equivalent
to f with N | A' and B' = B (mod 2*N) with A' minimal.
"""
A,B,C = f
u = ZZ(-B/(2*A/N))
v2 = ZZ( abs(D) / (2*A / N)^2 )
raise NotImplementedError
}}}
We make a list of the binary quadratic forms of discriminant $D$
that have first entry divisible by $N$.
{{{
def fill_list(D, N, b):
b = ZZ(b)
R = ZZ((b^2 - D)/(4*N))
L = []
Lprime = []
for d in divisors(R):
f = BinaryQF([d*N,b,R/d])
fr = f.reduced()
if not fr in Lprime:
Lprime.append(fr)
L.append(f)
return L
}}}
{{{
b = S[0]; print b
L = fill_list(D, N, b); L
///
17
[37x^2 + 17xy + 2y^2 ]
}}}
{{{
def tau(f):
A,B,C = f
return (-B + sqrt(D))/(2*A)
}}}
{{{
v = [CDF(tau(f)) for f in L]
v
///
[-0.22972972973 + 0.0357533960955*I]
}}}
{{{
e = gp(E)
}}}
{{{
E2 = E.change_ring(CDF)
}}}
{{{
def phi(E, t, prec):
"""
Map a point t in the upper half plan to a point on the complex torus
representation of the elliptic curve, using prec terms of the
modular parametrization.
"""
q = CDF(exp(2*pi*I*t))
a = E.anlist(prec)
return sum( (a[n]/n) * q^n for n in xrange(1,prec))
}}}
{{{
phi(E, v[0], 100)
///
0.929592715295 - 1.22569469101*I
}}}
{{{
phi(E, v[0], 200)
///
0.929592715279 - 1.225694691*I
}}}
{{{
def topoint(E, z):
"""
Given a point in the complex lattice representation of E, return
a point on the Weierstrass model of E. The curve E should be defined
over a complex floating point field.
"""
e = gp(E)
w = list(e.ellztopoint(z))
K = E.base_field()
return E.point([K(repr(w[0])), K(repr(w[1])), K(1)], check=False)
}}}
{{{
topoint(E2, phi(E, v[0], 200))
///
(-6.3956744052e-12 - 6.60496957297e-12*I : 6.3956744052e-12 + 6.60496957289e-12*I : 1.0)
}}}
Next we try discriminant -67
In this example, the class number is still 1, but the Gross-Zagier
formula predicts an index of 6.
{{{
D = -67
K. = QuadraticField(D); K
///
Number Field in a with defining polynomial x^2 + 67
}}}
{{{
K.class_number()
///
1
}}}
{{{
srs = Mod(D, 4*N).sqrt(all=True); srs
///
[9, 65, 83, 139]
}}}
{{{
S = list(set([Mod(a, 2*N) for a in srs])); S
///
[9, 65]
}}}
{{{
b = S[0]; print b
L = fill_list(D, N, b); L
///
9
[37x^2 + 9xy + y^2 ]
}}}
{{{
v = [CDF(tau(f)) for f in L]; v
///
[-0.121621621622 + 0.110612875296*I]
}}}
{{{
phi(E, v[0], 100)
///
0.409361000751 - 2.44021449129e-12*I
}}}
{{{
topoint(E2, phi(E, v[0], 100))
///
(6.00000000002 + 7.07662202477e-11*I : -15.0000000001 - 2.61102950569e-10*I : 1.0)
}}}
{{{
-topoint(E2, phi(E, v[0], 100))
///
(6.00000000002 + 7.07662202477e-11*I : 14.0000000001 + 2.61102950569e-10*I : 1)
}}}
{{{
b = S[1]; print b
L = fill_list(D, N, b); L
v = [CDF(tau(f)) for f in L]; v
///
65
[-0.878378378378 + 0.110612875296*I]
}}}
{{{
topoint(E2, phi(E, v[0], 100))
///
(6.00000000002 - 7.07662202477e-11*I : -15.0000000001 + 2.61102950569e-10*I : 1.0)
}}}
{{{
-topoint(E2, phi(E, v[0], 100))
///
(6.00000000002 - 7.07662202477e-11*I : 14.0000000001 - 2.61102950569e-10*I : 1)
}}}
Next we consider discriminant -40, which has class number 2 and Heegner
index 2
{{{
D = -40
K. = QuadraticField(D); K
///
Number Field in a with defining polynomial x^2 + 40
}}}
{{{
K.class_number()
///
2
}}}
{{{
srs = Mod(D, 4*N).sqrt(all=True); srs
///
[16, 58, 90, 132]
}}}
{{{
S = list(set([Mod(a, 2*N) for a in srs])); S
///
[16, 58]
}}}
We use the first $b$, as usual. Note that there are two representatives
since the class group has order 2.
{{{
b = S[0]; print b
L = fill_list(D, N, b); L
///
16
[37x^2 + 16xy + 2y^2 , 74x^2 + 16xy + y^2 ]
}}}
These are the corresponding Heegner points in the upper half plane.
{{{
v = [CDF(tau(f)) for f in L]
v
///
[-0.216216216216 + 0.0854669637883*I, -0.108108108108 + 0.0427334818942*I]
}}}
We compute their images in the complex torus representation of the elliptic curve.
{{{
e = [phi(E, z, 100) for z in v]
e
///
[0.567136607832 - 0.572182654882*I, 0.567136607828 + 0.572182654884*I]
}}}
Their sum in the torus is a point that must map to the Heegner point
$y_K$ on the
Weierstrass equation, which is defined over $\mathbf{Q}$.
{{{
w = e[0] + e[1]
}}}
This point is not exactly the point in the table above. In fact
it is negative of that point.
{{{
topoint(E2, w)
///
(1.0 - 1.51800794158e-12*I : -1.0 + 3.03601588315e-12*I : 1.0)
}}}
{{{
-topoint(E2, w)
///
(1.0 - 1.51800794158e-12*I : 2.33657537763e-12 - 3.03601588315e-12*I : 1)
}}}
Hilbert Class Field
We could also find the image of $y_1 \in E(H)$ point on the Weierstrass equation,
which is a point defined over the Hilber class field of $K$, which is a
quadratic extension of the field $K$.
{{{
y1 = topoint(E2, e[0])
}}}
{{{
y1
///
(9.04203384991e-12 + 1.41421356237*I : 0.99999999999 - 1.41421356238*I : 1.0)
}}}
{{{
CDF(sqrt(2))
///
1.41421356237
}}}
As you can see, the point $y_1$ is defined over $\mathbf{Q}(\sqrt{-2})$.
We have ``magically'' constructed the Hilbert class field of $K$ -- it's
$K(\sqrt{-2})$.
{{{
}}}
Finally we consider discriminant -71, which has class number 7 and Heegner
index 1
{{{
D = -71
K. = QuadraticField(D); K
///
Number Field in a with defining polynomial x^2 + 71
}}}
{{{
K.class_number()
///
7
}}}
{{{
srs = Mod(D, 4*N).sqrt(all=True); srs
///
[15, 59, 89, 133]
}}}
{{{
S = list(set([Mod(a, 2*N) for a in srs])); S
///
[59, 15]
}}}
We use the first $b$, as usual. Note that there are two representatives
since the class group has order 2.
{{{
b = S[0]; print b
L = fill_list(D, N, b); L
///
59
[37x^2 + 59xy + 24y^2 , 74x^2 + 59xy + 12y^2 , 111x^2 + 59xy + 8y^2 , 148x^2 + 59xy + 6y^2 , 222x^2 + 59xy + 4y^2 , 444x^2 + 59xy + 2y^2 , 888x^2 + 59xy + y^2 ]
}}}
These are the corresponding Heegner points in the upper half plane.
{{{
v = [CDF(tau(f)) for f in L]; v
///
[-0.797297297297 + 0.113866888827*I, -0.398648648649 + 0.0569334444134*I, -0.265765765766 + 0.0379556296089*I, -0.199324324324 + 0.0284667222067*I, -0.132882882883 + 0.0189778148045*I, -0.0664414414414 + 0.00948890740223*I, -0.0332207207207 + 0.00474445370111*I]
}}}
We compute their images in the complex torus representation of the elliptic curve.
{{{
e = [phi(E, z, 1000) for z in v]; e
///
[0.436338629693 + 0.391421370491*I, -1.08506665883 - 0.563889008593*I, 0.302913505562 - 1.22569469099*I, 1.52920424182 - 0.558966284143*I, 1.52920424182 + 0.558966284147*I, -1.08506665882 + 0.563889008594*I, 0.436338629696 - 0.391421370488*I]
}}}
Their sum in the torus is a point that must map to the Heegner point
$y_K$ on the
Weierstrass equation, which is defined over $\mathbf{Q}$.
{{{
w = sum(e); w
///
2.06386593094 - 1.22569469098*I
}}}
This point is not exactly the point in the table above. In fact
it is negative of that point.
{{{
topoint(E2, w)
///
(6.56395542695e-12 - 1.33950304269e-11*I : -0.999999999993 - 1.33950304271e-11*I : 1.0)
}}}
{{{
-topoint(E2, w)
///
(6.56395542695e-12 - 1.33950304269e-11*I : -6.56397158849e-12 + 1.33950304271e-11*I : 1)
}}}
{{{
y1 = topoint(E2, e[0]); y1
///
(0.326282033345 - 2.82634364479*I : 2.48918514468 + 4.09829358564*I : 1.0)
}}}
{{{
RR. = PolynomialRing(CDF)
f = prod(x - topoint(E2, e[i])[0] for i in xrange(len(e)))
f
///
1.0*x^7 + (-2.00000000003 - 9.62474544508e-12*I)*x^6 + (9.00000000011 - 1.00197627972e-10*I)*x^5 + (-10.0000000003 + 1.69921854365e-11*I)*x^4 + (-0.999999999939 + 1.66228364407e-10*I)*x^3 + (8.00000000023 - 1.35994326911e-10*I)*x^2 + (-5.0000000002 + 3.67623709252e-11*I)*x + 1.00000000005 - 2.89157586764e-12*I
}}}
{{{
SS. = PolynomialRing(QQ)
f = x^7 - 2*x^6 + 9*x^5 - 10*x^4 -x^3 + 8*x^2 -5*x + 1
factor(f.discriminant())
///
-1 * 71^3
}}}
{{{
}}}