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Adding methods .x() and .y() for EllipticCurvePoint #37092

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Feb 2, 2024
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Adding methods .x() and .y() for EllipticCurvePoint #37092

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8923a6a
Adding methods .x() and .y() for EllipticCurvePoint
r98inver Jan 18, 2024
ac527fb
Replaced `== 1` with the faster `.is_one()` method for `.x()`, `.y()`…
r98inver Jan 21, 2024
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2 changes: 1 addition & 1 deletion src/sage/schemes/elliptic_curves/ell_generic.py
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Original file line number Diff line number Diff line change
Expand Up @@ -619,7 +619,7 @@ def _reduce_point(self, R, p):

sage: E = EllipticCurve([1,-1,0,94,9])
sage: R = E([0,3]) + 5*E([8,31])
sage: factor(R.xy()[0].denominator())
sage: factor(R.x().denominator())
2^2 * 11^2 * 1457253032371^2

Since 11 is a factor of the denominator, this point corresponds to the
Expand Down
64 changes: 55 additions & 9 deletions src/sage/schemes/elliptic_curves/ell_point.py
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Expand Up @@ -773,11 +773,57 @@ def xy(self):
...
ZeroDivisionError: rational division by zero
"""
if self[2] == 1:
if self[2].is_one():
return self[0], self[1]
else:
return self[0]/self[2], self[1]/self[2]

def x(self):
"""
Return the `x` coordinate of this point, as an element of the base field.
If this is the point at infinity, a :class:`ZeroDivisionError` is raised.

EXAMPLES::

sage: E = EllipticCurve('389a')
sage: P = E([-1,1])
sage: P.x()
-1
sage: Q = E(0); Q
(0 : 1 : 0)
sage: Q.x()
Traceback (most recent call last):
...
ZeroDivisionError: rational division by zero
"""
if self[2].is_one():
return self[0]
else:
return self[0]/self[2]

def y(self):
"""
Return the `y` coordinate of this point, as an element of the base field.
If this is the point at infinity, a :class:`ZeroDivisionError` is raised.

EXAMPLES::

sage: E = EllipticCurve('389a')
sage: P = E([-1,1])
sage: P.y()
1
sage: Q = E(0); Q
(0 : 1 : 0)
sage: Q.y()
Traceback (most recent call last):
...
ZeroDivisionError: rational division by zero
"""
if self[2].is_one():
return self[1]
else:
return self[1]/self[2]

def is_divisible_by(self, m):
"""
Return True if there exists a point `Q` defined over the same
Expand Down Expand Up @@ -1473,7 +1519,7 @@ def _miller_(self, Q, n):
sage: Fx.<b> = GF((2,(4*5)))
sage: Ex = EllipticCurve(Fx, [0,0,1,1,1])
sage: phi = Hom(F,Fx)(F.gen().minpoly().roots(Fx)[0][0])
sage: Px = Ex(phi(P.xy()[0]), phi(P.xy()[1]))
sage: Px = Ex(phi(P.x()), phi(P.y()))
sage: Qx = Ex(b^19 + b^18 + b^16 + b^12 + b^10 + b^9 + b^8 + b^5 + b^3 + 1,
....: b^18 + b^13 + b^10 + b^8 + b^5 + b^4 + b^3 + b)
sage: Px._miller_(Qx,41) == b^17 + b^13 + b^12 + b^9 + b^8 + b^6 + b^4 + 1
Expand Down Expand Up @@ -1658,7 +1704,7 @@ def weil_pairing(self, Q, n, algorithm=None):
sage: Fx.<b> = GF((2, 4*5))
sage: Ex = EllipticCurve(Fx, [0,0,1,1,1])
sage: phi = Hom(F, Fx)(F.gen().minpoly().roots(Fx)[0][0])
sage: Px = Ex(phi(P.xy()[0]), phi(P.xy()[1]))
sage: Px = Ex(phi(P.x()), phi(P.y()))
sage: O = Ex(0)
sage: Qx = Ex(b^19 + b^18 + b^16 + b^12 + b^10 + b^9 + b^8 + b^5 + b^3 + 1,
....: b^18 + b^13 + b^10 + b^8 + b^5 + b^4 + b^3 + b)
Expand Down Expand Up @@ -1898,7 +1944,7 @@ def tate_pairing(self, Q, n, k, q=None):
sage: Fx.<b> = GF((2,4*5))
sage: Ex = EllipticCurve(Fx,[0,0,1,1,1])
sage: phi = Hom(F, Fx)(F.gen().minpoly().roots(Fx)[0][0])
sage: Px = Ex(phi(P.xy()[0]), phi(P.xy()[1]))
sage: Px = Ex(phi(P.x()), phi(P.y()))
sage: Qx = Ex(b^19 + b^18 + b^16 + b^12 + b^10 + b^9 + b^8 + b^5 + b^3 + 1,
....: b^18 + b^13 + b^10 + b^8 + b^5 + b^4 + b^3 + b)
sage: Px.tate_pairing(Qx, n=41, k=4)
Expand Down Expand Up @@ -2070,7 +2116,7 @@ def ate_pairing(self, Q, n, k, t, q=None):
sage: Fx.<b> = GF(q^k)
sage: Ex = EllipticCurve(Fx, [0,0,1,1,1])
sage: phi = Hom(F, Fx)(F.gen().minpoly().roots(Fx)[0][0])
sage: Px = Ex(phi(P.xy()[0]), phi(P.xy()[1]))
sage: Px = Ex(phi(P.x()), phi(P.y()))
sage: Qx = Ex(b^19+b^18+b^16+b^12+b^10+b^9+b^8+b^5+b^3+1,
....: b^18+b^13+b^10+b^8+b^5+b^4+b^3+b)
sage: Qx = Ex(Qx[0]^q, Qx[1]^q) - Qx # ensure Qx is in ker(pi - q)
Expand Down Expand Up @@ -3862,14 +3908,14 @@ def padic_elliptic_logarithm(self,Q, p):
for k in range(0,p):
Eqp = EllipticCurve(Qp(p, 2), [ ZZ(t) + k * p for t in E.a_invariants() ])

P_Qps = Eqp.lift_x(ZZ(self.xy()[0]), all=True)
P_Qps = Eqp.lift_x(ZZ(self.x()), all=True)
for P_Qp in P_Qps:
if F(P_Qp.xy()[1]) == self.xy()[1]:
if F(P_Qp.y()) == self.y():
break

Q_Qps = Eqp.lift_x(ZZ(Q.xy()[0]), all=True)
Q_Qps = Eqp.lift_x(ZZ(Q.x()), all=True)
for Q_Qp in Q_Qps:
if F(Q_Qp.xy()[1]) == Q.xy()[1]:
if F(Q_Qp.y()) == Q.y():
break

pP = p * P_Qp
Expand Down
2 changes: 1 addition & 1 deletion src/sage/schemes/elliptic_curves/height.py
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Original file line number Diff line number Diff line change
Expand Up @@ -1177,7 +1177,7 @@ def psi(self, xi, v):
3.51086196882538
sage: L(P) / L.real_period()
0.867385122699931
sage: xP = v(P.xy()[0])
sage: xP = v(P.x())
sage: H = E.height_function()
sage: H.psi(xP, v)
0.867385122699931
Expand Down
2 changes: 1 addition & 1 deletion src/sage/schemes/elliptic_curves/hom_sum.py
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Expand Up @@ -306,7 +306,7 @@ def to_isogeny_chain(self):
from sage.groups.generic import multiples
from sage.misc.misc_c import prod
x = polygen(Kl.base_ring())
poly = prod(x - T.xy()[0] for T in multiples(Kl, l//2, Kl))
poly = prod(x - T.x() for T in multiples(Kl, l//2, Kl))
poly = poly.change_ring(self.base_ring())

psi = phi.codomain().isogeny(poly)
Expand Down
26 changes: 13 additions & 13 deletions src/sage/schemes/elliptic_curves/hom_velusqrt.py
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Expand Up @@ -260,7 +260,7 @@ class FastEllipticPolynomial:
Fast elliptic polynomial prod(Z - x(i*P) for i in range(1,n,2)) with n = 19, P = (4 : 35 : 1)
sage: hP(7)
19
sage: prod(7 - (i*P).xy()[0] for i in range(1,P.order(),2))
sage: prod(7 - (i*P).x() for i in range(1,P.order(),2))
19

Passing `Q` changes the index set::
Expand All @@ -269,7 +269,7 @@ class FastEllipticPolynomial:
sage: hPQ = FastEllipticPolynomial(E, P.order(), P, Q)
sage: hPQ(7)
58
sage: prod(7 - (Q+i*P).xy()[0] for i in range(P.order()))
sage: prod(7 - (Q+i*P).x() for i in range(P.order()))
58

The call syntax has an optional keyword argument ``derivative``, which
Expand All @@ -279,7 +279,7 @@ class FastEllipticPolynomial:
sage: hP(7, derivative=True)
(19, 15)
sage: R.<Z> = E.base_field()[]
sage: HP = prod(Z - (i*P).xy()[0] for i in range(1,P.order(),2))
sage: HP = prod(Z - (i*P).x() for i in range(1,P.order(),2))
sage: HP
Z^9 + 16*Z^8 + 57*Z^7 + 6*Z^6 + 45*Z^5 + 31*Z^4 + 46*Z^3 + 10*Z^2 + 28*Z + 41
sage: HP(7)
Expand All @@ -292,7 +292,7 @@ class FastEllipticPolynomial:
sage: hPQ(7, derivative=True)
(58, 62)
sage: R.<Z> = E.base_field()[]
sage: HPQ = prod(Z - (Q+i*P).xy()[0] for i in range(P.order()))
sage: HPQ = prod(Z - (Q+i*P).x() for i in range(P.order()))
sage: HPQ
Z^19 + 53*Z^18 + 67*Z^17 + 39*Z^16 + 56*Z^15 + 32*Z^14 + 44*Z^13 + 6*Z^12 + 27*Z^11 + 29*Z^10 + 38*Z^9 + 48*Z^8 + 38*Z^7 + 43*Z^6 + 21*Z^5 + 25*Z^4 + 33*Z^3 + 49*Z^2 + 60*Z
sage: HPQ(7)
Expand Down Expand Up @@ -342,9 +342,9 @@ def __init__(self, E, n, P, Q=None):
)

I, J, K = IJK
xI = (R.xy()[0] for R in _points_range(I, P, Q))
xJ = [R.xy()[0] for R in _points_range(J, P )]
xK = (R.xy()[0] for R in _points_range(K, P, Q))
xI = (R.x() for R in _points_range(I, P, Q))
xJ = [R.x() for R in _points_range(J, P )]
xK = (R.x() for R in _points_range(K, P, Q))

self.hItree = ProductTree(Z - xi for xi in xI)

Expand Down Expand Up @@ -766,7 +766,7 @@ def _raw_eval(self, x, y=None):
sage: phi._raw_codomain
Elliptic Curve defined by y^2 = x^3 + ... over Finite Field of size 65537
sage: Q = E(42, 15860)
sage: phi._raw_eval(Q.xy()[0])
sage: phi._raw_eval(Q.x())
11958
sage: phi._raw_eval(*Q.xy())
(11958, 42770)
Expand Down Expand Up @@ -794,8 +794,8 @@ def _raw_eval(self, x, y=None):
h0, h0d = self._h0(x, derivative=True)
h1, h1d = self._h1(x, derivative=True)

# assert h0 == prod(x - ( i*self._P).xy()[0] for i in range(1,self._P.order(),2))
# assert h1 == prod(x - (self._Q+i*self._P).xy()[0] for i in range( self._P.order() ))
# assert h0 == prod(x - ( i*self._P).x() for i in range(1,self._P.order(),2))
# assert h1 == prod(x - (self._Q+i*self._P).x() for i in range( self._P.order() ))

if not h0:
return ()
Expand All @@ -805,8 +805,8 @@ def _raw_eval(self, x, y=None):
if y is None:
return xx

# assert h0d == sum(prod(x - ( i*self._P).xy()[0] for i in range(1,self._P.order(),2) if i!=j) for j in range(1,self._P.order(),2))
# assert h1d == sum(prod(x - (self._Q+i*self._P).xy()[0] for i in range( self._P.order() ) if i!=j) for j in range( self._P.order() ))
# assert h0d == sum(prod(x - ( i*self._P).x() for i in range(1,self._P.order(),2) if i!=j) for j in range(1,self._P.order(),2))
# assert h1d == sum(prod(x - (self._Q+i*self._P).x() for i in range( self._P.order() ) if i!=j) for j in range( self._P.order() ))

yy = y * (h1d - 2 * h1 / h0 * h0d) / h0**2

Expand Down Expand Up @@ -1026,7 +1026,7 @@ def kernel_polynomial(self):
x^15 + 21562*x^14 + 8571*x^13 + 20029*x^12 + 1775*x^11 + 60402*x^10 + 17481*x^9 + 46543*x^8 + 46519*x^7 + 18590*x^6 + 36554*x^5 + 36499*x^4 + 48857*x^3 + 3066*x^2 + 23264*x + 53937
sage: h == E.isogeny(K).kernel_polynomial()
True
sage: h(K.xy()[0])
sage: h(K.x())
0

TESTS::
Expand Down
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