Magma V2.14-7     Mon Mar 10 2008 10:56:50 on stan     [Seed = 1171747939]
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> Attach("twocovdesc.m");
> 
> Qx<x>:=PolynomialRing(Rationals());
> 
> ///////////////////////////////////////////////////////////////////////////////
> //Example 7.1
> 
> C:=HyperellipticCurve(2*x^6+x+2);
> Hk,AtoHk:=TwoCoverDescent(C);
> #Hk;
0
> 
> ///////////////////////////////////////////////////////////////////////////////
> //Example 7.2
> 
> C:=HyperellipticCurve(-x^6+2*x^5+3*x^4-x^3+x^2+x-3);
> Hk,AtoHk:=TwoCoverDescent(C);
> #Hk;
0
> 
> ///////////////////////////////////////////////////////////////////////////////
> //Example 8.1
> 
> f:=2*x^6+x^4+3*x^2-2;
> C:=HyperellipticCurve(f);
> Hk,AtoHk:=TwoCoverDescent(C);
> A<theta>:=Domain(AtoHk);
> deltas:={-1-theta,1-theta};
> {AtoHk(d): d in deltas} eq Hk;
true
> L<alpha>:=NumberField(x^2+x+2);
> LX<X>:=PolynomialRing(L);
> g:=(X^2-1/2)*(X^2-alpha);
> h:=2*(X^2+alpha+1);
> g*h eq Evaluate(f,X);
true
> 
> LTHETA<THETA>:=quo<LX|g>;
> j:=hom<A->LTHETA|THETA>;
> {Norm(j(delta)):delta in deltas} eq {1/2*(-alpha + 1)};
true
> E:=HyperellipticCurve( (1-alpha)*(2*X^2-1)*(X^2-alpha));
> P1:=ProjectiveSpace(Rationals(),1);
> EtoP1:=map<E->P1|[E.1,E.3]>;
> 
> P0:=E![1,alpha-1];
> Eprime,EtoEprime:=EllipticCurve(E,P0);
> Etilde:=EllipticCurve(X^3+(1-alpha)*X^2+(2-9*alpha)*X+(16-2*alpha));
> EprimeToEtilde:=Isomorphism(Eprime,Etilde);
> 
> EtoEtilde:=EtoEprime*EprimeToEtilde;
> 
> EtildeToP1:=Expand(Inverse(EtoEtilde)*EtoP1);
> 
> success,MWgrp,MWmap:=PseudoMordellWeilGroup(Etilde);
> assert success;
> MWgrp;
Abelian Group isomorphic to Z/2 + Z
Defined on 2 generators
Relations:
    2*MWgrp.1 = 0
> N,V,R:=Chabauty(MWmap,EtildeToP1,5);
> N eq #V;
true
> R;
4
> {EvaluateByPowerSeries(EtildeToP1,MWmap(g)):g in V};
{ (1 : 1), (-1 : 1) }
> RationalPoints(C,1);
{@ (1 : 2 : 1), (1 : -2 : 1) @}
> RationalPoints(C,-1);
{@ (-1 : 2 : 1), (-1 : -2 : 1) @}
> 
> ///////////////////////////////////////////////////////////////////////////////
> // Example 9.1
> 
> k<alpha>:=NumberField(x^2-2);
> kX<X>:=PolynomialRing(k);
> E:=EllipticCurve(X^3+(2-2*alpha)*X+(2-9*alpha));
> Hk,AtoHk:=TwoCoverDescent(HyperellipticCurve(E));
> A<theta>:=Domain(AtoHk);
> delta:=theta^2+(8-4*alpha)*theta+13-6*alpha;
> two:=MultiplicationByMMap(E,2);
> mu,tor:=CasselsMap(two);
> Sel2,mp:=SelmerGroup(two);
> B:=Domain(mp);
> Modulus(A) eq Modulus(B);
true
> AtoB:=hom<A->B|B.1>;
> {mp(AtoB(d@@AtoHk)):d in Hk} eq {a: a in Sel2};
true
> D:=Domain(tor(AtoB(delta)));
> C:=Quartic(D);
> HkC:=TwoCoverDescent(C);
> #HkC;
0
> 
> 

Total time: 74.769 seconds, Total memory usage: 36.61MB