Changeset 7683:f0960b48c972

Timestamp:

11/20/07 03:12:24 (6 years ago)

Author:

Robert Bradshaw <robertwb@…>

Branch:

default

Message:

Isomorphisms between different Weierstrass models of an elliptic curve

Location:

sage

Files:

• rings/number_field/number_field_element.pyx (modified) (1 diff)
• schemes/elliptic_curves/ell_generic.py (modified) (2 diffs)
• schemes/elliptic_curves/ell_rational_field.py (modified) (2 diffs)
• schemes/elliptic_curves/weierstrass_morphism.py (added)

sage/rings/number_field/number_field_element.pyx

@@ -678 +678 @@
         else:
             raise ValueError, "%s not a square in %s"%(self, self._parent)
+            
+    def nth_root(self, n, all=False):
+        """
+        Return an nth root of self in the given number field. 
+        
+        EXAMPLES: 
+            sage: K.<a> = NumberField(x^4-7)
+            sage: K(7).nth_root(2)
+            a^2
+            sage: K((a-3)^5).nth_root(5)
+            a - 3
+
+        ALGORITHM: 
+            Use Pari to factor $x^n$ - \code{self} in K. 
+        """
+        R = sage.rings.polynomial.polynomial_ring.PolynomialRing(self.number_field(), 't')
+        if not self:
+            return [self] if all else self
+        f = (R.gen(0) << (n-1)) - self
+        roots = f.roots()
+        if all:
+            return [r[0] for r in roots]
+        elif len(roots) > 0:
+            return roots[0][0]
+        else:
+            raise ValueError, "%s not a %s-th root in %s"%(self, n, self._parent)
+        
 
     cdef void _reduce_c_(self):

sage/schemes/elliptic_curves/ell_generic.py

@@ -56 +56 @@
 import constructor
 import formal_group
+import weierstrass_morphism
 
 factor = arith.factor
@@ -1282 +1283 @@
 
     division_polynomial = torsion_polynomial
-            
+
+    def isomporphism_to(self, other):
+        """
+        Given another weierstrass model \code{other} of self, return a morphism
+        from self to \code{other}.
+        
+        If the curves in question are not isomorphic, raise a ValueError
+        
+        EXAMPLES: 
+            sage: E = EllipticCurve('37a')
+            sage: F = E.weierstrass_model()
+            sage: w = E.isomporphism_to(F); w
+            Generic morphism:
+              From: Abelian group of points on Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
+              To:   Abelian group of points on Elliptic Curve defined by y^2  = x^3 - x + 1/4 over Rational Field
+            sage: P = E(0,-1,1)
+            sage: w(P)
+            (0 : -1/2 : 1)
+            sage: w(5*P)
+            (1/4 : 1/8 : 1)
+            sage: 5*w(P)
+            (1/4 : 1/8 : 1)
+            sage: 120*w(P) == w(120*P)
+            True
+            
+          We can also handle injections to different base rings: 
+              sage: K.<a> = NumberField(x^3-7)
+              sage: E.isomporphism_to(E.change_ring(K))
+              Generic morphism:
+                From: Abelian group of points on Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
+                To:   Abelian group of points on Elliptic Curve defined by y^2 + y = x^3 + (-1)*x over Number Field in a with defining polynomial x^3 - 7
+        """
+        return weierstrass_morphism.WeierstrassIsomorphism(self, other)
+    
+    def is_isomorphic(self, other):
+        """
+        Returns whether or not self is isomorphic to other, i.e. they define the same curve over 
+        the same basering. 
+        
+        EXAMPLES: 
+            sage: E = EllipticCurve('389a')
+            sage: F = E.change_weierstrass_model([2,3,4,5]); F
+            Elliptic Curve defined by y^2 - 8*x*y + 22*y = x^3 - 21*x^2 + 59*x over Rational Field
+            sage: E.is_isomorphic(F)
+            True
+            sage: E.is_isomorphic(F.change_ring(CC))
+            False
+        """
+        if not is_EllipticCurve(other):
+            return False
+        elif self.base_ring() != other.base_ring():
+            return False
+        else:
+            try:
+                phi = self.isomporphism_to(other)
+                return True
+            except ValueError:
+                return False
+        
+    def change_weierstrass_model(self, *urst):
+        """
+        Return a new Weierstrass model of self under the transformation (on points)
+            $$ (x,y) \mapsto (x',y') = (u^2*x+r , u^3*y + s*u^2*x' + t) $$
+        """
+        if isinstance(urst[0], (tuple, list)):
+            urst = urst[0]
+        u, r, s, t = urst
+        a1, a2, a3, a4, a6 = self.ainvs()
+        b1 = a1*u - 2*s
+        b3 = a3*u**3 - a1*r*u - 2*t + 2*r*s
+        b2 = a2*u**2 + a1*s*u - s**2 - 3*r
+        b4 = a4*u**4 + a3*s*u**3 - 2*a2*r*u**2 + a1*t*u - 2*a1*r*s*u - 2*s*t + 2*r*s**2 + 3*r**2
+        b6 = a6*u**6 - a4*r*u**4 + a3*t*u**3 - a3*r*s*u**3 + a2*r**2*u**2 - a1*r*t*u + a1*r**2*s*u - t**2 + 2*r*s*t - r**2*s**2 - r**3
+        return constructor.EllipticCurve(self.base_ring(), [b1,b2,b3,b4,b6])
+        
     def weierstrass_model(self):
         """

sage/schemes/elliptic_curves/ell_rational_field.py

@@ -1409 +1409 @@
         return True
 
-    def is_isomorphic(self, E):
-        if not isinstance(E, EllipticCurve_rational_field):
-            raise TypeError, "E (=%s) must be an elliptic curve over the rational numbers"%E
-        return E.minimal_model() == self.minimal_model()
-
     def kodaira_type(self, p):
         """
@@ -1574 +1569 @@
         F = self.minimal_model()
         return EllipticCurve_number_field.weierstrass_model(F)
+        
+    def local_integral_model(self,p):
+        r"""
+        Return a model of self which is integral at the prime $p$
+        
+        EXAMPLES:
+            sage: E=EllipticCurve([0, 0, 1/216, -7/1296, 1/7776])
+            sage: E.local_integral_model(2)
+             Elliptic Curve defined by y^2 + 1/27*y = x^3 - 7/81*x + 2/243 over Rational Field
+            sage: E.local_integral_model(3)
+             Elliptic Curve defined by y^2 + 1/8*y = x^3 - 7/16*x + 3/32 over Rational Field
+            sage: E.local_integral_model(2).local_integral_model(3) == EllipticCurve('5077a1')
+            True
+        """
+        ai = self.a_invariants()
+        e  = min([(ai[i].valuation(p)/[1,2,3,4,6][i]) for i in range(5)]).floor()
+        return constructor.EllipticCurve([ai[i]/p**(e*[1,2,3,4,6][i]) for i in range(5)])
+
+    def global_integral_model(self):
+        r"""
+        Return a model of self which is integral at all primes
+        
+        EXAMPLES:
+            sage: E=EllipticCurve([0, 0, 1/216, -7/1296, 1/7776])
+            sage: E.global_integral_model() == EllipticCurve('5077a1')
+            True
+        """
+        ai = self.a_invariants()
+        for a in ai:
+            if not a.is_integral():
+      ### Is there really no prime_factors() function?
+               pj=[fi[0] for fi in a.denom().factor()]
+               for p in pj:
+                  e  = min([(ai[i].valuation(p)/[1,2,3,4,6][i]) for i in range(5)]).floor()
+                  ai = [ai[i]/p**(e*[1,2,3,4,6][i]) for i in range(5)]
+            return constructor.EllipticCurve(ai)
+
+    integral_model = global_integral_model
 
     def integral_weierstrass_model(self):