// Magma code (for each group separately) for the Theorem 3.11 and Table 1
// each portion of the code goes through all quadratic fields, for 0<d<100, and determines whether that group is a possible torsion subgroup or not over all mentioned fields


// Z/11Z
S:=[n : n in [2..100] | SquareFree(n) eq n];   // set of all non-negative squarefree integers <100
_<x>:=PolynomialRing(Rationals());
E:=EllipticCurve([0,-1,-1,0,0]);
RankBounds(E);
for d in S do
	printf "d=%o\n", d;
    E1:=QuadraticTwist(E,d);
    r1,r2:=RankBounds(E1);
    r1; r2;
    if r2 eq 0 then
        E2:=BaseChange(E,QuadraticField(d));
        TorsionSubgroup(E2);
    end if;
end for;



// Z/14Z
S:=[n : n in [2..100] | SquareFree(n) eq n]; 
_<x>:=PolynomialRing(Rationals());
E:=EllipticCurve([1,0,1,-1,0]);
RankBounds(E);
for d in S do
	printf "d=%o\n", d;
    E1:=QuadraticTwist(E,d);
    r1,r2:=RankBounds(E1);
    r1; r2;
    if r2 eq 0 then
        E2:=BaseChange(E,QuadraticField(d));
        TorsionSubgroup(E2);
    end if;
end for;



// Z/15Z
S:=[n : n in [2..100] | SquareFree(n) eq n];  
_<x>:=PolynomialRing(Rationals());
E:=EllipticCurve([1,1,1,0,0]);
RankBounds(E);
for d in S do
	printf "d=%o\n", d;
    E1:=QuadraticTwist(E,d);
    r1,r2:=RankBounds(E1);
    r1; r2;
    if r2 eq 0 then
        E2:=BaseChange(E,QuadraticField(d));
        TorsionSubgroup(E2);
    end if;
end for;



// Z/2Z x Z/10Z
S:=[n : n in [2..100] | SquareFree(n) eq n];  
_<x>:=PolynomialRing(Rationals());
E:=EllipticCurve([0,1,0,-1,0]);
RankBounds(E);
for d in S do
	printf "d=%o\n", d;
    E1:=QuadraticTwist(E,d);
    r1,r2:=RankBounds(E1);
    r1; r2;
    if r2 eq 0 then
        E2:=BaseChange(E,QuadraticField(d));
        TorsionSubgroup(E2);
    end if;
end for;



// Z/2Z x Z/12Z
S:=[n : n in [2..100] | SquareFree(n) eq n];  
_<x>:=PolynomialRing(Rationals());
E:=EllipticCurve([0,-1,0,1,0]);
RankBounds(E);
for d in S do
	printf "d=%o\n", d;
    E1:=QuadraticTwist(E,d);
    r1,r2:=RankBounds(E1);
    r1; r2;
    if r2 eq 0 then
        E2:=BaseChange(E,QuadraticField(d));
        TorsionSubgroup(E2);
    end if;
end for;



// Z/13Z
// In the set S we put only d such that d≡1(mod 8) (see Theorem 3.6)
S:=[17,33,41,57,65,73,89,97]; 
_<x>:=PolynomialRing(Rationals());
C:=HyperellipticCurve(x^6-2*x^5+x^4-2*x^3+6*x^2-4*x+1);
J:=Jacobian(C);
RankBounds(J);
TorsionSubgroup(J);
Points(J:Bound:=100);
for d in S do
    printf "d=%o\n", d;
    C1:=QuadraticTwist(C,d);
    J1:=Jacobian(C1);
    r1,r2:=RankBounds(J1);
    r1; r2;
    TorsionSubgroup(J1);
    if r2 ne 0 then
        C2:=ChangeRing(C,QuadraticField(d));
        Points(C2:Bound:=200);
    end if;
end for;



//Z/16Z
S:=[n : n in [2..100] | SquareFree(n) eq n];  
_<x>:=PolynomialRing(Rationals());
J:=JOne(16);
L:=LSeries(J);
IsZeroAt(L,1);     // this is false iff rank(J_1(16)(Q))=0
tr, p:=NewModularHyperellipticCurve(ModularSymbols(J));
C1:=HyperellipticCurve(p);

C:=HyperellipticCurve(x*(x^2+1)*(x^2+2*x-1));
J1:=Jacobian(C);
TorsionSubgroup(J1);
Points(J1:Bound:=100);

for d in S do
    printf "d=%o\n", d;
    C2:=QuadraticTwist(C1,d);
    J2:=Jacobian(C2);
    r1, r2:=RankBounds(J2);
    r1; r2;
    if r2 eq 0 then			// if the rank of the Jacobian is 0 then we compute the torsion subgroup (as in Proposition 3.9)
        C2:=QuadraticTwist(C,d);
        J2:=Jacobian(C2);
        TorsionSubgroup(J2);
        Points(J2:Bound:=100);
        J3:=BaseChange(J1,QuadraticField(d));
        TwoTorsionSubgroup(J3);
    else				// if the rank of the Jacobian is not 0, then we cannot use the methods described in Proposition 3.9, so we are searching for the points on the curve
        C3:=ChangeRing(C,QuadraticField(d));
        Points(C3:Bound:=1000);		// we can put whichever bound we want
    end if;
end for;



// Z/18Z
// In the set S we put only d such that d≡1(mod 8) and d≠2(mod 3) (see Theorem 3.7)
S:=[33,57,73,97]; 
_<x>:=PolynomialRing(Rationals());
C:=HyperellipticCurve(x^6+2*x^5+5*x^4+10*x^3+10*x^2+4*x+1);
J:=Jacobian(C);
RankBounds(J);
TorsionSubgroup(J);
Points(J:Bound:=100);
for d in S do
    printf "d=%o\n", d;
    C1:=QuadraticTwist(C,d);
    J1:=Jacobian(C1);
    r1,r2:=RankBounds(J1);
    r1; r2;
    TorsionSubgroup(J1);
    if r2 ne 0 then
        C2:=ChangeRing(C,QuadraticField(d));
        Points(C2:Bound:=200);
    end if;
end for;