def EllipticCurve_from_cubic(F, P):
r"""
Construct an elliptic curve from a ternary cubic with a rational point.
INPUT:
- ``F`` -- a homogeneous cubic in three variables with rational
coefficients (either as a polynomial ring element or as a
string) defining a smooth plane cubic curve.
- ``P`` -- a 3-tuple `(x,y,z)` defining a projective point on the
curve `F=0`.
OUTPUT:
(elliptic curve) An elliptic curve (in minimal Weierstrass form)
isomorphic to the curve `F=0`.
.. note::
USES MAGMA - This function will not work on computers that
do not have magma installed.
TO DO: implement this without using MAGMA.
For a more general version, see the function
``EllipticCurve_from_plane_curve()``.
EXAMPLES:
First we find that the Fermat cubic is isomorphic to the curve
with Cremona label 27a1::
sage: E = EllipticCurve_from_cubic('x^3 + y^3 + z^3', [1,-1,0]) # optional - magma
sage: E # optional - magma
Elliptic Curve defined by y^2 + y = x^3 - 7 over Rational Field
sage: E.cremona_label() # optional - magma
'27a1'
Next we find the minimal model and conductor of the Jacobian of the
Selmer curve.
::
sage: E = EllipticCurve_from_cubic('u^3 + v^3 + 60*w^3', [1,-1,0]) # optional - magma
sage: E # optional - magma
Elliptic Curve defined by y^2 = x^3 - 24300 over Rational Field
sage: E.conductor() # optional - magma
24300
"""
from sage.interfaces.all import magma
cmd = "P<%s,%s,%s> := ProjectivePlane(RationalField());" % SR(
F).variables()
magma.eval(cmd)
cmd = 'aInvariants(MinimalModel(EllipticCurve(Curve(Scheme(P, %s)),P!%s)));' % (
F, P)
s = magma.eval(cmd)
return EllipticCurve(rings.RationalField(), eval(s))
def EllipticCurve_from_c4c6(c4, c6):
"""
Return an elliptic curve with given `c_4` and
`c_6` invariants.
EXAMPLES::
sage: E = EllipticCurve_from_c4c6(17, -2005)
sage: E
Elliptic Curve defined by y^2 = x^3 - 17/48*x + 2005/864 over Rational Field
sage: E.c_invariants()
(17, -2005)
"""
try:
K = c4.parent()
except AttributeError:
K = rings.RationalField()
if K not in _Fields:
K = K.fraction_field()
return EllipticCurve([-K(c4) / K(48), -K(c6) / K(864)])
def coefficients_from_j(j, minimal_twist=True):
"""
Return Weierstrass coefficients `(a_1, a_2, a_3, a_4, a_6)` for an
elliptic curve with given `j`-invariant.
INPUT:
See :func:`EllipticCurve_from_j`.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.constructor import coefficients_from_j
sage: coefficients_from_j(0)
[0, 0, 1, 0, 0]
sage: coefficients_from_j(1728)
[0, 0, 0, -1, 0]
sage: coefficients_from_j(1)
[1, 0, 0, 36, 3455]
The ``minimal_twist`` parameter (ignored except over `\\QQ` and
True by default) controls whether or not a minimal twist is
computed::
sage: coefficients_from_j(100)
[0, 1, 0, 3392, 307888]
sage: coefficients_from_j(100, minimal_twist=False)
[0, 0, 0, 488400, -530076800]
"""
try:
K = j.parent()
except AttributeError:
K = rings.RationalField()
if K not in _Fields:
K = K.fraction_field()
char = K.characteristic()
if char == 2:
if j == 0:
return Sequence([0, 0, 1, 0, 0], universe=K)
else:
return Sequence([1, 0, 0, 0, 1 / j], universe=K)
if char == 3:
if j == 0:
return Sequence([0, 0, 0, 1, 0], universe=K)
else:
return Sequence([0, j, 0, 0, -j**2], universe=K)
if K is rings.RationalField():
# we construct the minimal twist, i.e. the curve with minimal
# conductor with this j_invariant:
if j == 0:
return Sequence([0, 0, 1, 0, 0], universe=K) # 27a3
if j == 1728:
return Sequence([0, 0, 0, -1, 0], universe=K) # 32a2
if not minimal_twist:
k = j - 1728
return Sequence([0, 0, 0, -3 * j * k, -2 * j * k**2], universe=K)
n = j.numerator()
m = n - 1728 * j.denominator()
a4 = -3 * n * m
a6 = -2 * n * m**2
# Now E=[0,0,0,a4,a6] has j-invariant j=n/d
from sage.sets.set import Set
for p in Set(n.prime_divisors() + m.prime_divisors()):
e = min(a4.valuation(p) // 2, a6.valuation(p) // 3)
if e > 0:
p = p**e
a4 /= p**2
a6 /= p**3
# Now E=[0,0,0,a4,a6] is minimal at all p != 2,3
tw = [-1, 2, -2, 3, -3, 6, -6]
E1 = EllipticCurve([0, 0, 0, a4, a6])
Elist = [E1] + [E1.quadratic_twist(t) for t in tw]
Elist.sort(key=lambda E: E.conductor())
return Sequence(Elist[0].ainvs())
# defaults for all other fields:
if j == 0:
return Sequence([0, 0, 0, 0, 1], universe=K)
if j == 1728:
return Sequence([0, 0, 0, 1, 0], universe=K)
k = j - 1728
return Sequence([0, 0, 0, -3 * j * k, -2 * j * k**2], universe=K)
def EllipticCurve_from_j(j, minimal_twist=True):
"""
Return an elliptic curve with given `j`-invariant.
INPUT:
- ``j`` -- an element of some field.
- ``minimal_twist`` (boolean, default True) -- If True and ``j`` is in `\QQ`, the curve returned is a
minimal twist, i.e. has minimal conductor. If `j` is not in `\QQ` this parameter is ignored.
OUTPUT:
An elliptic curve with `j`-invariant `j`.
EXAMPLES::
sage: E = EllipticCurve_from_j(0); E; E.j_invariant(); E.label()
Elliptic Curve defined by y^2 + y = x^3 over Rational Field
0
'27a3'
sage: E = EllipticCurve_from_j(1728); E; E.j_invariant(); E.label()
Elliptic Curve defined by y^2 = x^3 - x over Rational Field
1728
'32a2'
sage: E = EllipticCurve_from_j(1); E; E.j_invariant()
Elliptic Curve defined by y^2 + x*y = x^3 + 36*x + 3455 over Rational Field
1
The ``minimal_twist`` parameter (ignored except over `\QQ` and
True by default) controls whether or not a minimal twist is
computed::
sage: EllipticCurve_from_j(100)
Elliptic Curve defined by y^2 = x^3 + x^2 + 3392*x + 307888 over Rational Field
sage: _.conductor()
33129800
sage: EllipticCurve_from_j(100, minimal_twist=False)
Elliptic Curve defined by y^2 = x^3 + 488400*x - 530076800 over Rational Field
sage: _.conductor()
298168200
Since computing the minimal twist requires factoring both `j` and
`j-1728` the following example would take a long time without
setting ``minimal_twist`` to False::
sage: E = EllipticCurve_from_j(2^256+1,minimal_twist=False)
sage: E.j_invariant() == 2^256+1
True
"""
try:
K = j.parent()
except AttributeError:
K = rings.RationalField()
if K not in _Fields:
K = K.fraction_field()
char = K.characteristic()
if char == 2:
if j == 0:
return EllipticCurve(K, [0, 0, 1, 0, 0])
else:
return EllipticCurve(K, [1, 0, 0, 0, 1 / j])
if char == 3:
if j == 0:
return EllipticCurve(K, [0, 0, 0, 1, 0])
else:
return EllipticCurve(K, [0, j, 0, 0, -j**2])
if K is rings.RationalField():
# we construct the minimal twist, i.e. the curve with minimal
# conductor with this j_invariant:
if j == 0:
return EllipticCurve(K, [0, 0, 1, 0, 0]) # 27a3
if j == 1728:
return EllipticCurve(K, [0, 0, 0, -1, 0]) # 32a2
if not minimal_twist:
k = j - 1728
return EllipticCurve(K, [0, 0, 0, -3 * j * k, -2 * j * k**2])
n = j.numerator()
m = n - 1728 * j.denominator()
a4 = -3 * n * m
a6 = -2 * n * m**2
# Now E=[0,0,0,a4,a6] has j-invariant j=n/d
from sage.sets.set import Set
for p in Set(n.prime_divisors() + m.prime_divisors()):
e = min(a4.valuation(p) // 2, a6.valuation(p) // 3)
if e > 0:
p = p**e
a4 /= p**2
a6 /= p**3
# Now E=[0,0,0,a4,a6] is minimal at all p != 2,3
tw = [-1, 2, -2, 3, -3, 6, -6]
E1 = EllipticCurve([0, 0, 0, a4, a6])
Elist = [E1] + [E1.quadratic_twist(t) for t in tw]
crv_cmp = lambda E, F: cmp(E.conductor(), F.conductor())
Elist.sort(cmp=crv_cmp)
return Elist[0]
# defaults for all other fields:
if j == 0:
return EllipticCurve(K, [0, 0, 0, 0, 1])
if j == 1728:
return EllipticCurve(K, [0, 0, 0, 1, 0])
k = j - 1728
return EllipticCurve(K, [0, 0, 0, -3 * j * k, -2 * j * k**2])
def EllipticCurve_from_plane_curve(C, P):
r"""
Construct an elliptic curve from a smooth plane cubic with a rational point.
INPUT:
- ``C`` -- a plane curve of genus one.
- ``P`` -- a 3-tuple `(x,y,z)` defining a projective point on the
curve ``C``.
OUTPUT:
(elliptic curve) An elliptic curve (in minimal Weierstrass form)
isomorphic to ``C``.
.. note::
USES MAGMA - This function will not work on computers that
do not have magma installed.
TO DO: implement this without using MAGMA.
EXAMPLES:
First we check that the Fermat cubic is isomorphic to the curve
with Cremona label '27a1'::
sage: x,y,z=PolynomialRing(QQ,3,'xyz').gens() # optional - magma
sage: C=Curve(x^3+y^3+z^3) # optional - magma
sage: P=C(1,-1,0) # optional - magma
sage: E=EllipticCurve_from_plane_curve(C,P) # optional - magma
sage: E # optional - magma
Elliptic Curve defined by y^2 + y = x^3 - 7 over Rational Field
sage: E.label() # optional - magma
'27a1'
Now we try a quartic example::
sage: u,v,w=PolynomialRing(QQ,3,'uvw').gens() # optional - magma
sage: C=Curve(u^4+u^2*v^2-w^4) # optional - magma
sage: P=C(1,0,1) # optional - magma
sage: E=EllipticCurve_from_plane_curve(C,P) # optional - magma
sage: E # optional - magma
Elliptic Curve defined by y^2 = x^3 + 4*x over Rational Field
sage: E.label() # optional - magma
'32a1'
"""
from sage.interfaces.all import magma
if C.genus() != 1:
raise TypeError, "The curve C must have genus 1"
elif P.parent() != C.point_set(C.base_ring()):
raise TypeError, "The point P must be on the curve C"
dp = C.defining_polynomial()
x, y, z = dp.parent().variable_names()
cmd = "PR<%s,%s,%s>:=ProjectivePlane(RationalField());" % (x, y, z)
magma.eval(cmd)
cmd = 'CC:=Curve(PR, %s);' % (dp)
magma.eval(cmd)
cmd = 'aInvariants(MinimalModel(EllipticCurve(CC,CC!%s)));' % (
[P[0], P[1], P[2]])
s = magma.eval(cmd)
return EllipticCurve(rings.RationalField(), eval(s))
def EllipticCurve_from_j(j):
"""
Return an elliptic curve with given `j`-invariant.
EXAMPLES::
sage: E = EllipticCurve_from_j(0); E; E.j_invariant(); E.label()
Elliptic Curve defined by y^2 + y = x^3 over Rational Field
0
'27a3'
sage: E = EllipticCurve_from_j(1728); E; E.j_invariant(); E.label()
Elliptic Curve defined by y^2 = x^3 - x over Rational Field
1728
'32a2'
sage: E = EllipticCurve_from_j(1); E; E.j_invariant()
Elliptic Curve defined by y^2 + x*y = x^3 + 36*x + 3455 over Rational Field
1
"""
try:
K = j.parent()
except AttributeError:
K = rings.RationalField()
if not rings.is_Field(K):
K = K.fraction_field()
char=K.characteristic()
if char==2:
if j == 0:
return EllipticCurve(K, [ 0, 0, 1, 0, 0 ])
else:
return EllipticCurve(K, [ 1, 0, 0, 0, 1/j ])
if char == 3:
if j==0:
return EllipticCurve(K, [ 0, 0, 0, 1, 0 ])
else:
return EllipticCurve(K, [ 0, j, 0, 0, -j**2 ])
if K is rings.RationalField():
# we construct the minimal twist, i.e. the curve with minimal
# conductor with this j_invariant:
if j == 0:
return EllipticCurve(K, [ 0, 0, 1, 0, 0 ]) # 27a3
if j == 1728:
return EllipticCurve(K, [ 0, 0, 0, -1, 0 ]) # 32a2
n = j.numerator()
m = n-1728*j.denominator()
a4 = -3*n*m
a6 = -2*n*m**2
# Now E=[0,0,0,a4,a6] has j-invariant j=n/d
from sage.sets.set import Set
for p in Set(n.prime_divisors()+m.prime_divisors()):
e = min(a4.valuation(p)//2,a6.valuation(p)//3)
if e>0:
p = p**e
a4 /= p**2
a6 /= p**3
# Now E=[0,0,0,a4,a6] is minimal at all p != 2,3
tw = [-1,2,-2,3,-3,6,-6]
E1 = EllipticCurve([0,0,0,a4,a6])
Elist = [E1] + [E1.quadratic_twist(t) for t in tw]
crv_cmp = lambda E,F: cmp(E.conductor(),F.conductor())
Elist.sort(cmp=crv_cmp)
return Elist[0]
# defaults for all other fields:
if j == 0:
return EllipticCurve(K, [ 0, 0, 0, 0, 1 ])
if j == 1728:
return EllipticCurve(K, [ 0, 0, 0, 1, 0 ])
k=j-1728
return EllipticCurve(K, [0,0,0,-3*j*k, -2*j*k**2])
