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#19689: unit scaling for Weierstrass models of elliptic curves over n…
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…umber fields
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@JohnCremona
JohnCremona committed Jan 5, 2016
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113 changes: 63 additions & 50 deletions src/sage/schemes/elliptic_curves/ell_number_field.py
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Original file line number Diff line number Diff line change
Expand Up @@ -916,14 +916,11 @@ def _scale_by_units(self):
OUTPUT:
A model for this elliptic curve, optimally scaled with repect
to scaling by units when the base field is real quadratic;
unchanged otherwise.
to scaling by units, with respect to the logarithmic embedding
of |c4|^(1/4)+|c6|^(1/6). No scaling by roots of unity is
carried out, so there is no change when the unit rank is 0.
.. note::
This is currently only implemented for real quadratic fields.
EXAMPLE::
EXAMPLES::
sage: K.<a> = NumberField(x^2-10)
sage: u = K.units()[0]
Expand All @@ -937,32 +934,48 @@ def _scale_by_units(self):
-77225139016967233228487820912*a - 244207331916752959911655344864)
sage: E1._scale_by_units().ainvs()
(0, 0, 0, 4536*a + 14148, -163728*a - 474336)
A totally real cubic example::
sage: K.<a> = NumberField(x^3-x^2-6*x+5)
sage: E = EllipticCurve([a + 1, a^2 + a - 1, a + 1, 44*a^2 + a - 258, -215*a^2 + 53*a + 1340])
sage: u1, u2 = K.units()
sage: u = u1^2/u2^3
sage: E1 = E.scale_curve(u)
sage: E1._scale_by_units().ainvs() == E.ainvs()
True
A complex quartic example::
sage: K.<a> = CyclotomicField(5)
sage: E = EllipticCurve([a + 1, a^2 + a - 1, a + 1, 44*a^2 + a - 258, -215*a^2 + 53*a + 1340])
sage: u = K.units()[0]
sage: E1 = E.scale_curve(u^5)
sage: E1._scale_by_units().ainvs() == E.ainvs()
True
"""
K = self.base_field()
if not K.signature()==(2,0):
r1, r2 = K.signature()
if r1+r2==1: # unit rank is 0
return self

R = RealField(1000) # lower precision works badly!
embs = K.embeddings(R)
u = K.units()[0]
uv = [e(u).abs().log() for e in embs]
prec = 1000 # lower precision works badly!
embs = K.places(prec=prec)
degs = [1]*r1 + [2]*r2
fu = K.units()
from sage.matrix.all import Matrix
U = Matrix([[e(u).abs().log()*d for d,e in zip(degs,embs)] for u in fu])
A = U*U.transpose()
Ainv = A.inverse()

c4, c6 = self.c_invariants()
c4s = [e(c4) for e in embs]
c6s = [e(c6) for e in embs]
v = [(x4.abs()**(R.one()/4)+x6.abs()**(R.one()/6)).log() for x4,x6 in zip(c4s,c6s)]
kr = -(v[0]*uv[0]+v[1]*uv[1])/(uv[0]**2+uv[1]**2)
k1 = kr.floor()
k2 = kr.ceil()
nv1 = (v[0] + k1*uv[0])**2 + (v[1] + k1*uv[1])**2
nv2 = (v[0] + k2*uv[0])**2 + (v[1] + k2*uv[1])**2
if nv1 < nv2:
k=k1
else:
k=k2
return self.scale_curve(u**k)


from sage.modules.all import vector
v = vector([(x4.abs().nth_root(4)+x6.abs().nth_root(6)).log()*d for x4,x6,d in zip(c4s,c6s,degs)])
es = [e.round() for e in -Ainv*U*v]
u = prod([uj**ej for uj,ej in zip(fu,es)])
return self.scale_curve(u)

def local_data(self, P=None, proof=None, algorithm="pari", globally=False):
r"""
Expand Down Expand Up @@ -1890,7 +1903,7 @@ def global_minimal_model(self, proof = None, semi_global=False):
sage: K2.<v> = K1.extension(m^2-w+1)
sage: E = EllipticCurve([0*v,-432])
sage: E.global_minimal_model()
Elliptic Curve defined by y^2 + (v+w+1)*y = x^3 + ((6*w-10)*v+16*w+20) over Number Field in v with defining polynomial x^2 - w + 1 over its base field
Elliptic Curve defined by y^2 + y = x^3 over Number Field in v with defining polynomial x^2 - w + 1 over its base field
See :trac:`18662`: for fields of class number greater than 1,
even when global minimal models did exist, their computation
Expand Down Expand Up @@ -2707,10 +2720,10 @@ class :class:`EllipticCurveIsogeny` allowed composition. In
[1 2]
[2 1]
sage: [E.ainvs() for E in C]
[(0, 0, 0, -25762110*c^2 - 67447215, -154360009760*c^2 - 404119737340),
(0, 0, 0, 130763490*c^2 + 342343485, 1391590873420*c^3 + 3643232206680*c)]
[(0, 0, 0, 83490*c^2 - 147015, -64739840*c^2 - 84465260),
(0, 0, 0, -161535*c^2 + 70785, -62264180*c^3 + 6229080*c)]
sage: C.isogenies()[0][1]
Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + (-25762110*c^2-67447215)*x + (-154360009760*c^2-404119737340) over Number Field in c with defining polynomial x^4 + 3*x^2 + 1 to Elliptic Curve defined by y^2 = x^3 + (130763490*c^2+342343485)*x + (1391590873420*c^3+3643232206680*c) over Number Field in c with defining polynomial x^4 + 3*x^2 + 1
Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + (83490*c^2-147015)*x + (-64739840*c^2-84465260) over Number Field in c with defining polynomial x^4 + 3*x^2 + 1 to Elliptic Curve defined by y^2 = x^3 + (-161535*c^2+70785)*x + (-62264180*c^3+6229080*c) over Number Field in c with defining polynomial x^4 + 3*x^2 + 1
An example with CM by `\sqrt{-23}` (class number `3`)::
Expand Down Expand Up @@ -2746,11 +2759,11 @@ class number is only `3` is that the class also contains three
[-92, -23, -23]
sage: C.matrix()
[1 2 2 2 4 4]
[2 1 2 4 2 4]
[1 2 2 4 2 4]
[2 1 2 2 4 4]
[2 2 1 4 4 2]
[2 4 4 1 3 3]
[4 2 4 3 1 3]
[4 2 4 1 3 3]
[2 4 4 3 1 3]
[4 4 2 3 3 1]
The graph of this isogeny class has a shape which does not
Expand Down Expand Up @@ -2778,11 +2791,11 @@ class number is only `3` is that the class also contains three
sage: G = C.graph()
sage: G.adjacency_matrix()
[0 1 1 1 0 0]
[1 0 1 0 1 0]
[0 1 1 0 1 0]
[1 0 1 1 0 0]
[1 1 0 0 0 1]
[1 0 0 0 1 1]
[0 1 0 1 0 1]
[0 1 0 0 1 1]
[1 0 0 1 0 1]
[0 0 1 1 1 0]
To display the graph without any edge labels::
Expand Down Expand Up @@ -2838,10 +2851,10 @@ class number is only `3` is that the class also contains three
sage: C = E.isogeny_class(); len(C) # long time (~11s)
4
sage: C.matrix() # long time
[ 1 2 13 26]
[ 2 1 26 13]
[13 26 1 2]
[26 13 2 1]
[ 1 13 2 26]
[13 1 26 2]
[ 2 26 1 13]
[26 2 13 1]
sage: len(Set([EE.j_invariant() for EE in C.curves])) # long time
2
Expand All @@ -2864,20 +2877,20 @@ class number is only `3` is that the class also contains three
sage: CL.matrix() # long time
[1 2 3 3 5 5]
[2 1 5 5 3 3]
[3 5 1 3 5 2]
[3 5 3 1 2 5]
[5 3 5 2 1 3]
[5 3 2 5 3 1]
[3 5 1 3 2 5]
[3 5 3 1 5 2]
[5 3 2 5 1 3]
[5 3 5 2 3 1]
To see the array of binary quadratic forms::
sage: CL.qf_matrix() # long time
[[[1], [2, 0, 13], [3, -2, 9], [3, -2, 9], [5, -4, 6], [5, -4, 6]],
[[2, 0, 13], [1], [5, -4, 6], [5, -4, 6], [3, -2, 9], [3, -2, 9]],
[[3, -2, 9], [5, -4, 6], [1], [3, -2, 9], [5, -4, 6], [2, 0, 13]],
[[3, -2, 9], [5, -4, 6], [3, -2, 9], [1], [2, 0, 13], [5, -4, 6]],
[[5, -4, 6], [3, -2, 9], [5, -4, 6], [2, 0, 13], [1], [3, -2, 9]],
[[5, -4, 6], [3, -2, 9], [2, 0, 13], [5, -4, 6], [3, -2, 9], [1]]]
[[2, 0, 13], [1], [5, -4, 6], [5, -4, 6], [3, -2, 9], [3, -2, 9]],
[[3, -2, 9], [5, -4, 6], [1], [3, -2, 9], [2, 0, 13], [5, -4, 6]],
[[3, -2, 9], [5, -4, 6], [3, -2, 9], [1], [5, -4, 6], [2, 0, 13]],
[[5, -4, 6], [3, -2, 9], [2, 0, 13], [5, -4, 6], [1], [3, -2, 9]],
[[5, -4, 6], [3, -2, 9], [5, -4, 6], [2, 0, 13], [3, -2, 9], [1]]]
As in the non-CM case, the isogeny class may be visualized by
obtaining its graph and plotting it. Since there are more
Expand Down
6 changes: 3 additions & 3 deletions src/sage/schemes/elliptic_curves/isogeny_class.py
View file
Original file line number Diff line number Diff line change
Expand Up @@ -678,10 +678,10 @@ class :class:`EllipticCurveIsogeny` allowed composition. In
[1 2]
[2 1]
sage: [E.ainvs() for E in C]
[(0, 0, 0, -25762110*c^2 - 67447215, -154360009760*c^2 - 404119737340),
(0, 0, 0, 130763490*c^2 + 342343485, 1391590873420*c^3 + 3643232206680*c)]
[(0, 0, 0, 83490*c^2 - 147015, -64739840*c^2 - 84465260),
(0, 0, 0, -161535*c^2 + 70785, -62264180*c^3 + 6229080*c)]
sage: C.isogenies()[0][1]
Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + (-25762110*c^2-67447215)*x + (-154360009760*c^2-404119737340) over Number Field in c with defining polynomial x^4 + 3*x^2 + 1 to Elliptic Curve defined by y^2 = x^3 + (130763490*c^2+342343485)*x + (1391590873420*c^3+3643232206680*c) over Number Field in c with defining polynomial x^4 + 3*x^2 + 1
Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + (83490*c^2-147015)*x + (-64739840*c^2-84465260) over Number Field in c with defining polynomial x^4 + 3*x^2 + 1 to Elliptic Curve defined by y^2 = x^3 + (-161535*c^2+70785)*x + (-62264180*c^3+6229080*c) over Number Field in c with defining polynomial x^4 + 3*x^2 + 1
TESTS::
Expand Down
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