def _tate(self, proof = None, globally = False): r""" Tate's algorithm for an elliptic curve over a number field. Computes both local reduction data at a prime ideal and a local minimal model. The model is not required to be integral on input. If `P` is principal, uses a generator as uniformizer, so it will not affect integrality or minimality at other primes. If `P` is not principal, the minimal model returned will preserve integrality at other primes, but not minimality. The optional argument globally, when set to True, tells the algorithm to use the generator of the prime ideal if it is principal. Otherwise just any uniformizer will be used. .. note:: Called only by ``EllipticCurveLocalData.__init__()``. OUTPUT: (tuple) ``(Emin, p, val_disc, fp, KS, cp)`` where: - ``Emin`` (EllipticCurve) is a model (integral and) minimal at P - ``p`` (int) is the residue characteristic - ``val_disc`` (int) is the valuation of the local minimal discriminant - ``fp`` (int) is the valuation of the conductor - ``KS`` (string) is the Kodaira symbol - ``cp`` (int) is the Tamagawa number EXAMPLES (this raised a type error in sage prior to 4.4.4, see :trac:`7930`) :: sage: E = EllipticCurve('99d1') sage: R.<X> = QQ[] sage: K.<t> = NumberField(X^3 + X^2 - 2*X - 1) sage: L.<s> = NumberField(X^3 + X^2 - 36*X - 4) sage: EK = E.base_extend(K) sage: toK = EK.torsion_order() sage: da = EK.local_data() # indirect doctest sage: EL = E.base_extend(L) sage: da = EL.local_data() # indirect doctest EXAMPLES: The following example shows that the bug at :trac:`9324` is fixed:: sage: K.<a> = NumberField(x^2-x+6) sage: E = EllipticCurve([0,0,0,-53160*a-43995,-5067640*a+19402006]) sage: E.conductor() # indirect doctest Fractional ideal (18, 6*a) The following example shows that the bug at :trac:`9417` is fixed:: sage: K.<a> = NumberField(x^2+18*x+1) sage: E = EllipticCurve(K, [0, -36, 0, 320, 0]) sage: E.tamagawa_number(K.ideal(2)) 4 This is to show that the bug :trac: `11630` is fixed. (The computation of the class group would produce a warning):: sage: K.<t> = NumberField(x^7-2*x+177) sage: E = EllipticCurve([0,1,0,t,t]) sage: P = K.ideal(2,t^3 + t + 1) sage: E.local_data(P).kodaira_symbol() II """ E = self._curve P = self._prime K = E.base_ring() OK = K.maximal_order() t = verbose("Running Tate's algorithm with P = %s"%P, level=1) F = OK.residue_field(P) p = F.characteristic() # In case P is not principal we mostly use a uniformiser which # is globally integral (with positive valuation at some other # primes); for this to work, it is essential that we can # reduce (mod P) elements of K which are not integral (but are # P-integral). However, if the model is non-minimal and we # end up dividing a_i by pi^i then at that point we use a # uniformiser pi which has non-positive valuation at all other # primes, so that we can divide by it without losing # integrality at other primes. if globally: principal_flag = P.is_principal() else: principal_flag = False if (K is QQ) or principal_flag : pi = P.gens_reduced()[0] verbose("P is principal, generator pi = %s"%pi, t, 1) else: pi = K.uniformizer(P, 'positive') verbose("uniformizer pi = %s"%pi, t, 1) pi2 = pi*pi; pi3 = pi*pi2; pi4 = pi*pi3 pi_neg = None prime = pi if K is QQ else P pval = lambda x: x.valuation(prime) pdiv = lambda x: x.is_zero() or pval(x) > 0 # Since ResidueField is cached in a way that # does not care much about embeddings of number # fields, it can happen that F.p.ring() is different # from K. This is a problem: If F.p.ring() has no # embedding but K has, then there is no coercion # from F.p.ring().maximal_order() to K. But it is # no problem to do an explicit conversion in that # case (Simon King, trac ticket #8800). from sage.categories.pushout import pushout, CoercionException try: if hasattr(F.p.ring(), 'maximal_order'): # it is not ZZ _tmp_ = pushout(F.p.ring().maximal_order(),K) pinv = lambda x: F.lift(~F(x)) proot = lambda x,e: F.lift(F(x).nth_root(e, extend = False, all = True)[0]) preduce = lambda x: F.lift(F(x)) except CoercionException: # the pushout does not exist, we need conversion pinv = lambda x: K(F.lift(~F(x))) proot = lambda x,e: K(F.lift(F(x).nth_root(e, extend = False, all = True)[0])) preduce = lambda x: K(F.lift(F(x))) def _pquadroots(a, b, c): r""" Local function returning True iff `ax^2 + bx + c` has roots modulo `P` """ (a, b, c) = (F(a), F(b), F(c)) if a == 0: return (b != 0) or (c == 0) elif p == 2: return len(PolynomialRing(F, "x")([c,b,a]).roots()) > 0 else: return (b**2 - 4*a*c).is_square() def _pcubicroots(b, c, d): r""" Local function returning the number of roots of `x^3 + b*x^2 + c*x + d` modulo `P`, counting multiplicities """ return sum([rr[1] for rr in PolynomialRing(F, 'x')([F(d), F(c), F(b), F(1)]).roots()],0) if p == 2: halfmodp = OK(Integer(0)) else: halfmodp = pinv(Integer(2)) A = E.a_invariants() A = [0, A[0], A[1], A[2], A[3], 0, A[4]] indices = [1,2,3,4,6] if min([pval(a) for a in A if a != 0]) < 0: verbose("Non-integral model at P: valuations are %s; making integral"%([pval(a) for a in A if a != 0]), t, 1) e = 0 for i in range(7): if A[i] != 0: e = max(e, (-pval(A[i])/i).ceil()) pie = pi**e for i in range(7): if A[i] != 0: A[i] *= pie**i verbose("P-integral model is %s, with valuations %s"%([A[i] for i in indices], [pval(A[i]) for i in indices]), t, 1) split = None # only relevant for multiplicative reduction (a1, a2, a3, a4, a6) = (A[1], A[2], A[3], A[4], A[6]) while True: C = EllipticCurve([a1, a2, a3, a4, a6]); (b2, b4, b6, b8) = C.b_invariants() (c4, c6) = C.c_invariants() delta = C.discriminant() val_disc = pval(delta) if val_disc == 0: ## Good reduction already cp = 1 fp = 0 KS = KodairaSymbol("I0") break #return # Otherwise, we change coordinates so that p | a3, a4, a6 if p == 2: if pdiv(b2): r = proot(a4, 2) t = proot(((r + a2)*r + a4)*r + a6, 2) else: temp = pinv(a1) r = temp * a3 t = temp * (a4 + r*r) elif p == 3: if pdiv(b2): r = proot(-b6, 3) else: r = -pinv(b2) * b4 t = a1 * r + a3 else: if pdiv(c4): r = -pinv(12) * b2 else: r = -pinv(12*c4) * (c6 + b2 * c4) t = -halfmodp * (a1 * r + a3) r = preduce(r) t = preduce(t) verbose("Before first transform C = %s"%C) verbose("[a1,a2,a3,a4,a6] = %s"%([a1, a2, a3, a4, a6])) C = C.rst_transform(r, 0, t) (a1, a2, a3, a4, a6) = C.a_invariants() (b2, b4, b6, b8) = C.b_invariants() if min([pval(a) for a in (a1, a2, a3, a4, a6) if a != 0]) < 0: raise RuntimeError("Non-integral model after first transform!") verbose("After first transform %s\n, [a1,a2,a3,a4,a6] = %s\n, valuations = %s"%([r, 0, t], [a1, a2, a3, a4, a6], [pval(a1), pval(a2), pval(a3), pval(a4), pval(a6)]), t, 2) if pval(a3) == 0: raise RuntimeError("p does not divide a3 after first transform!") if pval(a4) == 0: raise RuntimeError("p does not divide a4 after first transform!") if pval(a6) == 0: raise RuntimeError("p does not divide a6 after first transform!") # Now we test for Types In, II, III, IV # NB the c invariants never change. if not pdiv(c4): # Multiplicative reduction: Type In (n = val_disc) split = False if _pquadroots(1, a1, -a2): cp = val_disc split = True elif Integer(2).divides(val_disc): cp = 2 else: cp = 1 KS = KodairaSymbol("I%s"%val_disc) fp = 1 break #return # Additive reduction if pval(a6) < 2: ## Type II KS = KodairaSymbol("II") fp = val_disc cp = 1 break #return if pval(b8) < 3: ## Type III KS = KodairaSymbol("III") fp = val_disc - 1 cp = 2 break #return if pval(b6) < 3: ## Type IV cp = 1 a3t = preduce(a3/pi) a6t = preduce(a6/pi2) if _pquadroots(1, a3t, -a6t): cp = 3 KS = KodairaSymbol("IV") fp = val_disc - 2 break #return # If our curve is none of these types, we change coords so that # p | a1, a2; p^2 | a3, a4; p^3 | a6 if p == 2: s = proot(a2, 2) # so s^2=a2 (mod pi) t = pi*proot(a6/pi2, 2) # so t^2=a6 (mod pi^3) elif p == 3: s = a1 # so a1'=2s+a1=3a1=0 (mod pi) t = a3 # so a3'=2t+a3=3a3=0 (mod pi^2) else: s = -a1*halfmodp # so a1'=2s+a1=0 (mod pi) t = -a3*halfmodp # so a3'=2t+a3=0 (mod pi^2) C = C.rst_transform(0, s, t) (a1, a2, a3, a4, a6) = C.a_invariants() (b2, b4, b6, b8) = C.b_invariants() verbose("After second transform %s\n[a1, a2, a3, a4, a6] = %s\nValuations: %s"%([0, s, t], [a1,a2,a3,a4,a6],[pval(a1),pval(a2),pval(a3),pval(a4),pval(a6)]), t, 2) if pval(a1) == 0: raise RuntimeError("p does not divide a1 after second transform!") if pval(a2) == 0: raise RuntimeError("p does not divide a2 after second transform!") if pval(a3) < 2: raise RuntimeError("p^2 does not divide a3 after second transform!") if pval(a4) < 2: raise RuntimeError("p^2 does not divide a4 after second transform!") if pval(a6) < 3: raise RuntimeError("p^3 does not divide a6 after second transform!") if min(pval(a1), pval(a2), pval(a3), pval(a4), pval(a6)) < 0: raise RuntimeError("Non-integral model after second transform!") # Analyze roots of the cubic T^3 + bT^2 + cT + d = 0 mod P, where # b = a2/p, c = a4/p^2, d = a6/p^3 b = preduce(a2/pi) c = preduce(a4/pi2) d = preduce(a6/pi3) bb = b*b cc = c*c bc = b*c w = 27*d*d - bb*cc + 4*b*bb*d - 18*bc*d + 4*c*cc x = 3*c - bb if pdiv(w): if pdiv(x): sw = 3 else: sw = 2 else: sw = 1 verbose("Analyzing roots of cubic T^3 + %s*T^2 + %s*T + %s, case %s"%(b, c, d, sw), t, 1) if sw == 1: ## Three distinct roots - Type I*0 verbose("Distinct roots", t, 1) KS = KodairaSymbol("I0*") cp = 1 + _pcubicroots(b, c, d) fp = val_disc - 4 break #return elif sw == 2: ## One double root - Type I*m for some m verbose("One double root", t, 1) ## Change coords so that the double root is T = 0 mod p if p == 2: r = proot(c, 2) elif p == 3: r = c * pinv(b) else: r = (bc - 9*d)*pinv(2*x) r = pi * preduce(r) C = C.rst_transform(r, 0, 0) (a1, a2, a3, a4, a6) = C.a_invariants() (b2, b4, b6, b8) = C.b_invariants() # The rest of this branch is just to compute cp, fp, KS. # We use pi to keep transforms integral. ix = 3; iy = 3; mx = pi2; my = mx while True: a2t = preduce(a2 / pi) a3t = preduce(a3 / my) a4t = preduce(a4 / (pi*mx)) a6t = preduce(a6 / (mx*my)) if pdiv(a3t*a3t + 4*a6t): if p == 2: t = my*proot(a6t, 2) else: t = my*preduce(-a3t*halfmodp) C = C.rst_transform(0, 0, t) (a1, a2, a3, a4, a6) = C.a_invariants() (b2, b4, b6, b8) = C.b_invariants() my *= pi iy += 1 a2t = preduce(a2 / pi) a3t = preduce(a3/my) a4t = preduce(a4/(pi*mx)) a6t = preduce(a6/(mx*my)) if pdiv(a4t*a4t - 4*a6t*a2t): if p == 2: r = mx*proot(a6t*pinv(a2t), 2) else: r = mx*preduce(-a4t*pinv(2*a2t)) C = C.rst_transform(r, 0, 0) (a1, a2, a3, a4, a6) = C.a_invariants() (b2, b4, b6, b8) = C.b_invariants() mx *= pi ix += 1 # and stay in loop else: if _pquadroots(a2t, a4t, a6t): cp = 4 else: cp = 2 break # exit loop else: if _pquadroots(1, a3t, -a6t): cp = 4 else: cp = 2 break KS = KodairaSymbol("I%s*"%(ix+iy-5)) fp = val_disc - ix - iy + 1 break #return else: # sw == 3 ## The cubic has a triple root verbose("Triple root", t, 1) ## First we change coordinates so that T = 0 mod p if p == 2: r = b elif p == 3: r = proot(-d, 3) else: r = -b * pinv(3) r = pi*preduce(r) C = C.rst_transform(r, 0, 0) (a1, a2, a3, a4, a6) = C.a_invariants() (b2, b4, b6, b8) = C.b_invariants() verbose("After third transform %s\n[a1,a2,a3,a4,a6] = %s\nValuations: %s"%([r,0,0],[a1,a2,a3,a4,a6],[pval(ai) for ai in [a1,a2,a3,a4,a6]]), t, 2) if min(pval(ai) for ai in [a1,a2,a3,a4,a6]) < 0: raise RuntimeError("Non-integral model after third transform!") if pval(a2) < 2 or pval(a4) < 3 or pval(a6) < 4: raise RuntimeError("Cubic after transform does not have a triple root at 0") a3t = preduce(a3/pi2) a6t = preduce(a6/pi4) # We test for Type IV* if not pdiv(a3t*a3t + 4*a6t): cp = 3 if _pquadroots(1, a3t, -a6t) else 1 KS = KodairaSymbol("IV*") fp = val_disc - 6 break #return # Now change coordinates so that p^3|a3, p^5|a6 if p==2: t = -pi2*proot(a6t, 2) else: t = pi2*preduce(-a3t*halfmodp) C = C.rst_transform(0, 0, t) (a1, a2, a3, a4, a6) = C.a_invariants() (b2, b4, b6, b8) = C.b_invariants() # We test for types III* and II* if pval(a4) < 4: ## Type III* KS = KodairaSymbol("III*") fp = val_disc - 7 cp = 2 break #return if pval(a6) < 6: ## Type II* KS = KodairaSymbol("II*") fp = val_disc - 8 cp = 1 break #return if pi_neg is None: if principal_flag: pi_neg = pi else: pi_neg = K.uniformizer(P, 'negative') pi_neg2 = pi_neg*pi_neg pi_neg3 = pi_neg*pi_neg2 pi_neg4 = pi_neg*pi_neg3 pi_neg6 = pi_neg4*pi_neg2 a1 /= pi_neg a2 /= pi_neg2 a3 /= pi_neg3 a4 /= pi_neg4 a6 /= pi_neg6 verbose("Non-minimal equation, dividing out...\nNew model is %s"%([a1, a2, a3, a4, a6]), t, 1) return (C, p, val_disc, fp, KS, cp, split)
def descend_to(self, K, f=None): r""" Given a subfield `K` and an elliptic curve self defined over a field `L`, this function determines whether there exists an elliptic curve over `K` which is isomorphic over `L` to self. If one exists, it finds it. INPUT: - `K` -- a subfield of the base field of self. - `f` -- an embedding of `K` into the base field of self. OUTPUT: Either an elliptic curve defined over `K` which is isomorphic to self or None if no such curve exists. .. NOTE:: This only works over number fields and QQ. EXAMPLES:: sage: E = EllipticCurve([1,2,3,4,5]) sage: E.descend_to(ZZ) Traceback (most recent call last): ... TypeError: Input must be a field. :: sage: F.<b> = QuadraticField(23) sage: G.<a> = F.extension(x^3+5) sage: E = EllipticCurve(j=1728*b).change_ring(G) sage: E.descend_to(F) Elliptic Curve defined by y^2 = x^3 + (8957952*b-206032896)*x + (-247669456896*b+474699792384) over Number Field in b with defining polynomial x^2 - 23 :: sage: L.<a> = NumberField(x^4 - 7) sage: K.<b> = NumberField(x^2 - 7) sage: E = EllipticCurve([a^6,0]) sage: E.descend_to(K) Elliptic Curve defined by y^2 = x^3 + 1296/49*b*x over Number Field in b with defining polynomial x^2 - 7 :: sage: K.<a> = QuadraticField(17) sage: E = EllipticCurve(j = 2*a) sage: print E.descend_to(QQ) None """ if not K.is_field(): raise TypeError, "Input must be a field." if self.base_field()==K: return self j = self.j_invariant() from sage.rings.all import QQ if K == QQ: f = QQ.embeddings(self.base_field())[0] if j in QQ: jbase = QQ(j) else: return None elif f == None: embeddings = K.embeddings(self.base_field()) if len(embeddings) == 0: raise TypeError, "Input must be a subfield of the base field of the curve." for g in embeddings: try: jbase = g.preimage(j) f = g break except StandardError: pass if f == None: return None else: try: jbase = f.preimage(j) except StandardError: return None E = EllipticCurve(j=jbase) E2 = EllipticCurve(self.base_field(), [f(a) for a in E.a_invariants()]) if jbase==0: d = self.is_sextic_twist(E2) if d == 1: return E if d == 0: return None Etwist = E2.sextic_twist(d) elif jbase==1728: d = self.is_quartic_twist(E2) if d == 1: return E if d == 0: return None Etwist = E2.quartic_twist(d) else: d = self.is_quadratic_twist(E2) if d == 1: return E if d == 0: return None Etwist = E2.quadratic_twist(d) if Etwist.is_isomorphic(self): try: Eout = EllipticCurve(K, [f.preimage(a) for a in Etwist.a_invariants()]) except StandardError: return None else: return Eout
def __init__(self, E=None, urst=None, F=None): r""" Constructor for WeierstrassIsomorphism class, INPUT: - ``E`` -- an EllipticCurve, or None (see below). - ``urst`` -- a 4-tuple `(u,r,s,t)`, or None (see below). - ``F`` -- an EllipticCurve, or None (see below). Given two Elliptic Curves ``E`` and ``F`` (represented by Weierstrass models as usual), and a transformation ``urst`` from ``E`` to ``F``, construct an isomorphism from ``E`` to ``F``. An exception is raised if ``urst(E)!=F``. At most one of ``E``, ``F``, ``urst`` can be None. If ``F==None`` then ``F`` is constructed as ``urst(E)``. If ``E==None`` then ``E`` is constructed as ``urst^-1(F)``. If ``urst==None`` then an isomorphism from ``E`` to ``F`` is constructed if possible, and an exception is raised if they are not isomorphic. Otherwise ``urst`` can be a tuple of length 4 or a object of type ``baseWI``. Users will not usually need to use this class directly, but instead use methods such as ``isomorphism`` of elliptic curves. EXAMPLES:: sage: from sage.schemes.elliptic_curves.weierstrass_morphism import * sage: WeierstrassIsomorphism(EllipticCurve([0,1,2,3,4]),(-1,2,3,4)) Generic morphism: From: Abelian group of points on Elliptic Curve defined by y^2 + 2*y = x^3 + x^2 + 3*x + 4 over Rational Field To: Abelian group of points on Elliptic Curve defined by y^2 - 6*x*y - 10*y = x^3 - 2*x^2 - 11*x - 2 over Rational Field Via: (u,r,s,t) = (-1, 2, 3, 4) sage: E=EllipticCurve([0,1,2,3,4]) sage: F=EllipticCurve(E.cremona_label()) sage: WeierstrassIsomorphism(E,None,F) Generic morphism: From: Abelian group of points on Elliptic Curve defined by y^2 + 2*y = x^3 + x^2 + 3*x + 4 over Rational Field To: Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x^2 + 3*x + 5 over Rational Field Via: (u,r,s,t) = (1, 0, 0, -1) sage: w=WeierstrassIsomorphism(None,(1,0,0,-1),F) sage: w._domain_curve==E True """ from ell_generic import is_EllipticCurve if E != None: if not is_EllipticCurve(E): raise ValueError, "First argument must be an elliptic curve or None" if F != None: if not is_EllipticCurve(F): raise ValueError, "Third argument must be an elliptic curve or None" if urst != None: if len(urst) != 4: raise ValueError, "Second argument must be [u,r,s,t] or None" if len([par for par in [E, urst, F] if par != None]) < 2: raise ValueError, "At most 1 argument can be None" if F == None: # easy case baseWI.__init__(self, *urst) F = EllipticCurve(baseWI.__call__(self, list(E.a_invariants()))) Morphism.__init__(self, Hom(E(0).parent(), F(0).parent())) self._domain_curve = E self._codomain_curve = F return if E == None: # easy case in reverse baseWI.__init__(self, *urst) inv_urst = baseWI.__invert__(self) E = EllipticCurve(baseWI.__call__(inv_urst, list(F.a_invariants()))) Morphism.__init__(self, Hom(E(0).parent(), F(0).parent())) self._domain_curve = E self._codomain_curve = F return if urst == None: # try to construct the morphism urst = isomorphisms(E, F, True) if urst == None: raise ValueError, "Elliptic curves not isomorphic." baseWI.__init__(self, *urst) Morphism.__init__(self, Hom(E(0).parent(), F(0).parent())) self._domain_curve = E self._codomain_curve = F return # none of the parameters is None: baseWI.__init__(self, *urst) if F != EllipticCurve(baseWI.__call__(self, list(E.a_invariants()))): raise ValueError, "second argument is not an isomorphism from first argument to third argument" else: Morphism.__init__(self, Hom(E(0).parent(), F(0).parent())) self._domain_curve = E self._codomain_curve = F return
def __init__(self, E=None, urst=None, F=None): r""" Constructor for WeierstrassIsomorphism class, INPUT: - ``E`` -- an EllipticCurve, or None (see below). - ``urst`` -- a 4-tuple `(u,r,s,t)`, or None (see below). - ``F`` -- an EllipticCurve, or None (see below). Given two Elliptic Curves ``E`` and ``F`` (represented by Weierstrass models as usual), and a transformation ``urst`` from ``E`` to ``F``, construct an isomorphism from ``E`` to ``F``. An exception is raised if ``urst(E)!=F``. At most one of ``E``, ``F``, ``urst`` can be None. If ``F==None`` then ``F`` is constructed as ``urst(E)``. If ``E==None`` then ``E`` is constructed as ``urst^-1(F)``. If ``urst==None`` then an isomorphism from ``E`` to ``F`` is constructed if possible, and an exception is raised if they are not isomorphic. Otherwise ``urst`` can be a tuple of length 4 or a object of type ``baseWI``. Users will not usually need to use this class directly, but instead use methods such as ``isomorphism`` of elliptic curves. EXAMPLES:: sage: from sage.schemes.elliptic_curves.weierstrass_morphism import * sage: WeierstrassIsomorphism(EllipticCurve([0,1,2,3,4]),(-1,2,3,4)) Generic morphism: From: Abelian group of points on Elliptic Curve defined by y^2 + 2*y = x^3 + x^2 + 3*x + 4 over Rational Field To: Abelian group of points on Elliptic Curve defined by y^2 - 6*x*y - 10*y = x^3 - 2*x^2 - 11*x - 2 over Rational Field Via: (u,r,s,t) = (-1, 2, 3, 4) sage: E=EllipticCurve([0,1,2,3,4]) sage: F=EllipticCurve(E.cremona_label()) sage: WeierstrassIsomorphism(E,None,F) Generic morphism: From: Abelian group of points on Elliptic Curve defined by y^2 + 2*y = x^3 + x^2 + 3*x + 4 over Rational Field To: Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x^2 + 3*x + 5 over Rational Field Via: (u,r,s,t) = (1, 0, 0, -1) sage: w=WeierstrassIsomorphism(None,(1,0,0,-1),F) sage: w._domain_curve==E True """ from ell_generic import is_EllipticCurve if E!=None: if not is_EllipticCurve(E): raise ValueError("First argument must be an elliptic curve or None") if F!=None: if not is_EllipticCurve(F): raise ValueError("Third argument must be an elliptic curve or None") if urst!=None: if len(urst)!=4: raise ValueError("Second argument must be [u,r,s,t] or None") if len([par for par in [E,urst,F] if par!=None])<2: raise ValueError("At most 1 argument can be None") if F==None: # easy case baseWI.__init__(self,*urst) F=EllipticCurve(baseWI.__call__(self,list(E.a_invariants()))) Morphism.__init__(self, Hom(E(0).parent(), F(0).parent())) self._domain_curve = E self._codomain_curve = F return if E==None: # easy case in reverse baseWI.__init__(self,*urst) inv_urst=baseWI.__invert__(self) E=EllipticCurve(baseWI.__call__(inv_urst,list(F.a_invariants()))) Morphism.__init__(self, Hom(E(0).parent(), F(0).parent())) self._domain_curve = E self._codomain_curve = F return if urst==None: # try to construct the morphism urst=isomorphisms(E,F,True) if urst==None: raise ValueError("Elliptic curves not isomorphic.") baseWI.__init__(self, *urst) Morphism.__init__(self, Hom(E(0).parent(), F(0).parent())) self._domain_curve = E self._codomain_curve = F return # none of the parameters is None: baseWI.__init__(self,*urst) if F!=EllipticCurve(baseWI.__call__(self,list(E.a_invariants()))): raise ValueError("second argument is not an isomorphism from first argument to third argument") else: Morphism.__init__(self, Hom(E(0).parent(), F(0).parent())) self._domain_curve = E self._codomain_curve = F return
def _tate(self, proof=None): r""" Tate's algorithm for an elliptic curve over a number field. Computes both local reduction data at a prime ideal and a local minimal model. The model is not required to be integral on input. If `P` is principal, uses a generator as uniformizer, so it will not affect integrality or minimality at other primes. If `P` is not principal, the minimal model returned will preserve integrality at other primes, but not minimality. .. note:: Called only by ``EllipticCurveLocalData.__init__()``. OUTPUT: (tuple) ``(Emin, p, val_disc, fp, KS, cp)`` where: - ``Emin`` (EllipticCurve) is a model (integral and) minimal at P - ``p`` (int) is the residue characteristic - ``val_disc`` (int) is the valuation of the local minimal discriminant - ``fp`` (int) is the valuation of the conductor - ``KS`` (string) is the Kodaira symbol - ``cp`` (int) is the Tamagawa number EXAMPLES (this raised a type error in sage prior to 4.4.4, see ticket #7930) :: sage: E = EllipticCurve('99d1') sage: R.<X> = QQ[] sage: K.<t> = NumberField(X^3 + X^2 - 2*X - 1) sage: L.<s> = NumberField(X^3 + X^2 - 36*X - 4) sage: EK = E.base_extend(K) sage: toK = EK.torsion_order() sage: da = EK.local_data() # indirect doctest sage: EL = E.base_extend(L) sage: da = EL.local_data() # indirect doctest EXAMPLES: The following example shows that the bug at #9324 is fixed:: sage: K.<a> = NumberField(x^2-x+6) sage: E = EllipticCurve([0,0,0,-53160*a-43995,-5067640*a+19402006]) sage: E.conductor() # indirect doctest Fractional ideal (18, 6*a) The following example shows that the bug at #9417 is fixed:: sage: K.<a> = NumberField(x^2+18*x+1) sage: E = EllipticCurve(K, [0, -36, 0, 320, 0]) sage: E.tamagawa_number(K.ideal(2)) 4 """ E = self._curve P = self._prime K = E.base_ring() OK = K.maximal_order() t = verbose("Running Tate's algorithm with P = %s" % P, level=1) F = OK.residue_field(P) p = F.characteristic() # In case P is not principal we mostly use a uniformiser which # is globally integral (with positive valuation at some other # primes); for this to work, it is essential that we can # reduce (mod P) elements of K which are not integral (but are # P-integral). However, if the model is non-minimal and we # end up dividing a_i by pi^i then at that point we use a # uniformiser pi which has non-positive valuation at all other # primes, so that we can divide by it without losing # integrality at other primes. principal_flag = P.is_principal() if principal_flag: pi = P.gens_reduced()[0] verbose("P is principal, generator pi = %s" % pi, t, 1) else: pi = K.uniformizer(P, 'positive') verbose("P is not principal, uniformizer pi = %s" % pi, t, 1) pi2 = pi * pi pi3 = pi * pi2 pi4 = pi * pi3 pi_neg = None prime = pi if K is QQ else P pval = lambda x: x.valuation(prime) pdiv = lambda x: x.is_zero() or pval(x) > 0 # Since ResidueField is cached in a way that # does not care much about embeddings of number # fields, it can happen that F.p.ring() is different # from K. This is a problem: If F.p.ring() has no # embedding but K has, then there is no coercion # from F.p.ring().maximal_order() to K. But it is # no problem to do an explicit conversion in that # case (Simon King, trac ticket #8800). from sage.categories.pushout import pushout, CoercionException try: if hasattr(F.p.ring(), 'maximal_order'): # it is not ZZ _tmp_ = pushout(F.p.ring().maximal_order(), K) pinv = lambda x: F.lift(~F(x)) proot = lambda x, e: F.lift( F(x).nth_root(e, extend=False, all=True)[0]) preduce = lambda x: F.lift(F(x)) except CoercionException: # the pushout does not exist, we need conversion pinv = lambda x: K(F.lift(~F(x))) proot = lambda x, e: K( F.lift(F(x).nth_root(e, extend=False, all=True)[0])) preduce = lambda x: K(F.lift(F(x))) def _pquadroots(a, b, c): r""" Local function returning True iff `ax^2 + bx + c` has roots modulo `P` """ (a, b, c) = (F(a), F(b), F(c)) if a == 0: return (b != 0) or (c == 0) elif p == 2: return len(PolynomialRing(F, "x")([c, b, a]).roots()) > 0 else: return (b**2 - 4 * a * c).is_square() def _pcubicroots(b, c, d): r""" Local function returning the number of roots of `x^3 + b*x^2 + c*x + d` modulo `P`, counting multiplicities """ return sum([ rr[1] for rr in PolynomialRing(F, 'x') ([F(d), F(c), F(b), F(1)]).roots() ], 0) if p == 2: halfmodp = OK(Integer(0)) else: halfmodp = pinv(Integer(2)) A = E.a_invariants() A = [0, A[0], A[1], A[2], A[3], 0, A[4]] indices = [1, 2, 3, 4, 6] if min([pval(a) for a in A if a != 0]) < 0: verbose( "Non-integral model at P: valuations are %s; making integral" % ([pval(a) for a in A if a != 0]), t, 1) e = 0 for i in range(7): if A[i] != 0: e = max(e, (-pval(A[i]) / i).ceil()) pie = pi**e for i in range(7): if A[i] != 0: A[i] *= pie**i verbose( "P-integral model is %s, with valuations %s" % ([A[i] for i in indices], [pval(A[i]) for i in indices]), t, 1) split = None # only relevant for multiplicative reduction (a1, a2, a3, a4, a6) = (A[1], A[2], A[3], A[4], A[6]) while True: C = EllipticCurve([a1, a2, a3, a4, a6]) (b2, b4, b6, b8) = C.b_invariants() (c4, c6) = C.c_invariants() delta = C.discriminant() val_disc = pval(delta) if val_disc == 0: ## Good reduction already cp = 1 fp = 0 KS = KodairaSymbol("I0") break #return # Otherwise, we change coordinates so that p | a3, a4, a6 if p == 2: if pdiv(b2): r = proot(a4, 2) t = proot(((r + a2) * r + a4) * r + a6, 2) else: temp = pinv(a1) r = temp * a3 t = temp * (a4 + r * r) elif p == 3: if pdiv(b2): r = proot(-b6, 3) else: r = -pinv(b2) * b4 t = a1 * r + a3 else: if pdiv(c4): r = -pinv(12) * b2 else: r = -pinv(12 * c4) * (c6 + b2 * c4) t = -halfmodp * (a1 * r + a3) r = preduce(r) t = preduce(t) verbose("Before first transform C = %s" % C) verbose("[a1,a2,a3,a4,a6] = %s" % ([a1, a2, a3, a4, a6])) C = C.rst_transform(r, 0, t) (a1, a2, a3, a4, a6) = C.a_invariants() (b2, b4, b6, b8) = C.b_invariants() if min([pval(a) for a in (a1, a2, a3, a4, a6) if a != 0]) < 0: raise RuntimeError, "Non-integral model after first transform!" verbose( "After first transform %s\n, [a1,a2,a3,a4,a6] = %s\n, valuations = %s" % ([r, 0, t], [a1, a2, a3, a4, a6], [pval(a1), pval(a2), pval(a3), pval(a4), pval(a6)]), t, 2) if pval(a3) == 0: raise RuntimeError, "p does not divide a3 after first transform!" if pval(a4) == 0: raise RuntimeError, "p does not divide a4 after first transform!" if pval(a6) == 0: raise RuntimeError, "p does not divide a6 after first transform!" # Now we test for Types In, II, III, IV # NB the c invariants never change. if not pdiv(c4): # Multiplicative reduction: Type In (n = val_disc) split = False if _pquadroots(1, a1, -a2): cp = val_disc split = True elif Integer(2).divides(val_disc): cp = 2 else: cp = 1 KS = KodairaSymbol("I%s" % val_disc) fp = 1 break #return # Additive reduction if pval(a6) < 2: ## Type II KS = KodairaSymbol("II") fp = val_disc cp = 1 break #return if pval(b8) < 3: ## Type III KS = KodairaSymbol("III") fp = val_disc - 1 cp = 2 break #return if pval(b6) < 3: ## Type IV cp = 1 a3t = preduce(a3 / pi) a6t = preduce(a6 / pi2) if _pquadroots(1, a3t, -a6t): cp = 3 KS = KodairaSymbol("IV") fp = val_disc - 2 break #return # If our curve is none of these types, we change coords so that # p | a1, a2; p^2 | a3, a4; p^3 | a6 if p == 2: s = proot(a2, 2) # so s^2=a2 (mod pi) t = pi * proot(a6 / pi2, 2) # so t^2=a6 (mod pi^3) elif p == 3: s = a1 # so a1'=2s+a1=3a1=0 (mod pi) t = a3 # so a3'=2t+a3=3a3=0 (mod pi^2) else: s = -a1 * halfmodp # so a1'=2s+a1=0 (mod pi) t = -a3 * halfmodp # so a3'=2t+a3=0 (mod pi^2) C = C.rst_transform(0, s, t) (a1, a2, a3, a4, a6) = C.a_invariants() (b2, b4, b6, b8) = C.b_invariants() verbose( "After second transform %s\n[a1, a2, a3, a4, a6] = %s\nValuations: %s" % ([0, s, t], [a1, a2, a3, a4, a6], [pval(a1), pval(a2), pval(a3), pval(a4), pval(a6)]), t, 2) if pval(a1) == 0: raise RuntimeError, "p does not divide a1 after second transform!" if pval(a2) == 0: raise RuntimeError, "p does not divide a2 after second transform!" if pval(a3) < 2: raise RuntimeError, "p^2 does not divide a3 after second transform!" if pval(a4) < 2: raise RuntimeError, "p^2 does not divide a4 after second transform!" if pval(a6) < 3: raise RuntimeError, "p^3 does not divide a6 after second transform!" if min(pval(a1), pval(a2), pval(a3), pval(a4), pval(a6)) < 0: raise RuntimeError, "Non-integral model after second transform!" # Analyze roots of the cubic T^3 + bT^2 + cT + d = 0 mod P, where # b = a2/p, c = a4/p^2, d = a6/p^3 b = preduce(a2 / pi) c = preduce(a4 / pi2) d = preduce(a6 / pi3) bb = b * b cc = c * c bc = b * c w = 27 * d * d - bb * cc + 4 * b * bb * d - 18 * bc * d + 4 * c * cc x = 3 * c - bb if pdiv(w): if pdiv(x): sw = 3 else: sw = 2 else: sw = 1 verbose( "Analyzing roots of cubic T^3 + %s*T^2 + %s*T + %s, case %s" % (b, c, d, sw), t, 1) if sw == 1: ## Three distinct roots - Type I*0 verbose("Distinct roots", t, 1) KS = KodairaSymbol("I0*") cp = 1 + _pcubicroots(b, c, d) fp = val_disc - 4 break #return elif sw == 2: ## One double root - Type I*m for some m verbose("One double root", t, 1) ## Change coords so that the double root is T = 0 mod p if p == 2: r = proot(c, 2) elif p == 3: r = c * pinv(b) else: r = (bc - 9 * d) * pinv(2 * x) r = pi * preduce(r) C = C.rst_transform(r, 0, 0) (a1, a2, a3, a4, a6) = C.a_invariants() (b2, b4, b6, b8) = C.b_invariants() # The rest of this branch is just to compute cp, fp, KS. # We use pi to keep transforms integral. ix = 3 iy = 3 mx = pi2 my = mx while True: a2t = preduce(a2 / pi) a3t = preduce(a3 / my) a4t = preduce(a4 / (pi * mx)) a6t = preduce(a6 / (mx * my)) if pdiv(a3t * a3t + 4 * a6t): if p == 2: t = my * proot(a6t, 2) else: t = my * preduce(-a3t * halfmodp) C = C.rst_transform(0, 0, t) (a1, a2, a3, a4, a6) = C.a_invariants() (b2, b4, b6, b8) = C.b_invariants() my *= pi iy += 1 a2t = preduce(a2 / pi) a3t = preduce(a3 / my) a4t = preduce(a4 / (pi * mx)) a6t = preduce(a6 / (mx * my)) if pdiv(a4t * a4t - 4 * a6t * a2t): if p == 2: r = mx * proot(a6t * pinv(a2t), 2) else: r = mx * preduce(-a4t * pinv(2 * a2t)) C = C.rst_transform(r, 0, 0) (a1, a2, a3, a4, a6) = C.a_invariants() (b2, b4, b6, b8) = C.b_invariants() mx *= pi ix += 1 # and stay in loop else: if _pquadroots(a2t, a4t, a6t): cp = 4 else: cp = 2 break # exit loop else: if _pquadroots(1, a3t, -a6t): cp = 4 else: cp = 2 break KS = KodairaSymbol("I%s*" % (ix + iy - 5)) fp = val_disc - ix - iy + 1 break #return else: # sw == 3 ## The cubic has a triple root verbose("Triple root", t, 1) ## First we change coordinates so that T = 0 mod p if p == 2: r = b elif p == 3: r = proot(-d, 3) else: r = -b * pinv(3) r = pi * preduce(r) C = C.rst_transform(r, 0, 0) (a1, a2, a3, a4, a6) = C.a_invariants() (b2, b4, b6, b8) = C.b_invariants() verbose( "After third transform %s\n[a1,a2,a3,a4,a6] = %s\nValuations: %s" % ([r, 0, 0], [a1, a2, a3, a4, a6], [pval(ai) for ai in [a1, a2, a3, a4, a6]]), t, 2) if min(pval(ai) for ai in [a1, a2, a3, a4, a6]) < 0: raise RuntimeError, "Non-integral model after third transform!" if pval(a2) < 2 or pval(a4) < 3 or pval(a6) < 4: raise RuntimeError, "Cubic after transform does not have a triple root at 0" a3t = preduce(a3 / pi2) a6t = preduce(a6 / pi4) # We test for Type IV* if not pdiv(a3t * a3t + 4 * a6t): cp = 3 if _pquadroots(1, a3t, -a6t) else 1 KS = KodairaSymbol("IV*") fp = val_disc - 6 break #return # Now change coordinates so that p^3|a3, p^5|a6 if p == 2: t = -pi2 * proot(a6t, 2) else: t = pi2 * preduce(-a3t * halfmodp) C = C.rst_transform(0, 0, t) (a1, a2, a3, a4, a6) = C.a_invariants() (b2, b4, b6, b8) = C.b_invariants() # We test for types III* and II* if pval(a4) < 4: ## Type III* KS = KodairaSymbol("III*") fp = val_disc - 7 cp = 2 break #return if pval(a6) < 6: ## Type II* KS = KodairaSymbol("II*") fp = val_disc - 8 cp = 1 break #return if pi_neg is None: if principal_flag: pi_neg = pi else: pi_neg = K.uniformizer(P, 'negative') pi_neg2 = pi_neg * pi_neg pi_neg3 = pi_neg * pi_neg2 pi_neg4 = pi_neg * pi_neg3 pi_neg6 = pi_neg4 * pi_neg2 a1 /= pi_neg a2 /= pi_neg2 a3 /= pi_neg3 a4 /= pi_neg4 a6 /= pi_neg6 verbose( "Non-minimal equation, dividing out...\nNew model is %s" % ([a1, a2, a3, a4, a6]), t, 1) return (C, p, val_disc, fp, KS, cp, split)
def descend_to(self, K, f=None): r""" Given a subfield `K` and an elliptic curve self defined over a field `L`, this function determines whether there exists an elliptic curve over `K` which is isomorphic over `L` to self. If one exists, it finds it. INPUT: - `K` -- a subfield of the base field of self. - `f` -- an embedding of `K` into the base field of self. OUTPUT: Either an elliptic curve defined over `K` which is isomorphic to self or None if no such curve exists. .. NOTE:: This only works over number fields and QQ. EXAMPLES:: sage: E = EllipticCurve([1,2,3,4,5]) sage: E.descend_to(ZZ) Traceback (most recent call last): ... TypeError: Input must be a field. :: sage: F.<b> = QuadraticField(23) sage: G.<a> = F.extension(x^3+5) sage: E = EllipticCurve(j=1728*b).change_ring(G) sage: E.descend_to(F) Elliptic Curve defined by y^2 = x^3 + (8957952*b-206032896)*x + (-247669456896*b+474699792384) over Number Field in b with defining polynomial x^2 - 23 :: sage: L.<a> = NumberField(x^4 - 7) sage: K.<b> = NumberField(x^2 - 7) sage: E = EllipticCurve([a^6,0]) sage: E.descend_to(K) Elliptic Curve defined by y^2 = x^3 + 1296/49*b*x over Number Field in b with defining polynomial x^2 - 7 :: sage: K.<a> = QuadraticField(17) sage: E = EllipticCurve(j = 2*a) sage: print E.descend_to(QQ) None """ if not K.is_field(): raise TypeError, "Input must be a field." if self.base_field() == K: return self j = self.j_invariant() from sage.rings.all import QQ if K == QQ: f = QQ.embeddings(self.base_field())[0] if j in QQ: jbase = QQ(j) else: return None elif f == None: embeddings = K.embeddings(self.base_field()) if len(embeddings) == 0: raise TypeError, "Input must be a subfield of the base field of the curve." for g in embeddings: try: jbase = g.preimage(j) f = g break except StandardError: pass if f == None: return None else: try: jbase = f.preimage(j) except StandardError: return None E = EllipticCurve(j=jbase) E2 = EllipticCurve(self.base_field(), [f(a) for a in E.a_invariants()]) if jbase == 0: d = self.is_sextic_twist(E2) if d == 1: return E if d == 0: return None Etwist = E2.sextic_twist(d) elif jbase == 1728: d = self.is_quartic_twist(E2) if d == 1: return E if d == 0: return None Etwist = E2.quartic_twist(d) else: d = self.is_quadratic_twist(E2) if d == 1: return E if d == 0: return None Etwist = E2.quadratic_twist(d) if Etwist.is_isomorphic(self): try: Eout = EllipticCurve( K, [f.preimage(a) for a in Etwist.a_invariants()]) except StandardError: return None else: return Eout