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

src/sage/schemes/elliptic_curves/ell_generic.py

@@ -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

src/sage/schemes/elliptic_curves/ell_point.py

@@ -778,6 +778,52 @@ def xy(self):
         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] == 1:
+            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] == 1:
+            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
@@ -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
@@ -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)
@@ -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)
@@ -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)
@@ -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

src/sage/schemes/elliptic_curves/height.py

@@ -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

src/sage/schemes/elliptic_curves/hom_sum.py

@@ -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)

src/sage/schemes/elliptic_curves/hom_velusqrt.py

@@ -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::
@@ -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
@@ -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)
@@ -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)
@@ -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)
 
@@ -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)
@@ -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 ()
@@ -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
 
@@ -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::