//
//
// code used in Proposition 2.2
//
//

K<u>:=PolynomialRing(Rationals());

A2:=(-4*(u^6+2*u^5+15*u^4-20*u^3+15*u^2+18*u+33)*(u-1)^2*(u+1)^2)/((u^2+3)^3*(u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31));
A4:=(64*(u^6+2*u^5+3*u^4-20*u^3+39*u^2+18*u+21)*(u-1)^6*(u+1)^6)/((u^2+3)^6*(u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31)^2);
A6:=(4096*(u-1)^12*(u+1)^12)/((u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31)^3*(u^2+3)^9);
E:=EllipticCurve([1,A2,0,A4,A6]);							// we create an elliptic curve given by the long Weierstrass model in section 2.4 of the paper D. Jeon, A. Schweizer "Torsion of rational elliptic curves over different types of cubic fields"

Factorization(Numerator(Discriminant(E)));						// gives the normalized factorization of the numerator of the discriminant
Factorization(Denominator(Discriminant(E)));						// gives the normalized factorization of the denominator of the discriminant
d_num:=Numerator(Discriminant(E));							// we have to check those because the factorization is normalized and we want to see whether there are any leading coefficients
d_den:=Denominator(Discriminant(E));
d_num;
d_den;

Factorization(Numerator(cInvariants(E)[1]));						// gives the normalized factorization of the numerator of the c_4-invariant
Factorization(Denominator(cInvariants(E)[1]));						// gives the normalized factorization of the denominator of the c_4-invariant
c4_num:=Numerator(cInvariants(E)[1]);							// we have to check those because the factorization is normalized and we want to see whether there are any leading coefficients
c4_den:=Denominator(cInvariants(E)[1]);	
c4_num;
c4_den;





//
// (1) ord_2(u)>0
//

K<u>:=PolynomialRing(Rationals());

A2:=(-4*(u^6+2*u^5+15*u^4-20*u^3+15*u^2+18*u+33)*(u-1)^2*(u+1)^2)/((u^2+3)^3*(u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31));
A4:=(64*(u^6+2*u^5+3*u^4-20*u^3+39*u^2+18*u+21)*(u-1)^6*(u+1)^6)/((u^2+3)^6*(u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31)^2);
A6:=(4096*(u-1)^12*(u+1)^12)/((u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31)^3*(u^2+3)^9);
E:=EllipticCurve([1,A2,0,A4,A6]);

Evaluate(d_num/2^14, 0);								// =1, so d_num/2^14 = 1 (mod 2)
Evaluate(d_den, 0);									// this is odd, so d_den =1 (mod 2)
Evaluate(c4_num, 0);									// this is odd, so c4_num =1 (mod 2)
Evaluate(c4_den, 0);									// this is odd, so c4_den =1 (mod 2)

aInvariants(E)[3];			// gives a_3, and we see that ord_2(a_3)>0
aInvariants(E)[4];			// gives a_4, and we see that ord_2(a_4)>0 (we have to look at the numerator)
aInvariants(E)[5];			// gives a_6, and we see that ord_2(a_6)>0 (we have to look at the numerator)
bInvariants(E)[1];			// gives b_2, and ord_2(b_2)=0, so the conditions of Lemma 2.1 are satisfied

aInvariants(E)[1];			// gives a_1=1
aInvariants(E)[2];			// gives a_2, and ord_2(a_2)>0 so we get the equation T^2+T=0 over F_2 in Lemma 2.1





//
// (2) ord_2(u)<0
//

K<m>:=PolynomialRing(Rationals());
u:=1/m;											// the substitution u-->1/m

A2:=(-4*(u^6+2*u^5+15*u^4-20*u^3+15*u^2+18*u+33)*(u-1)^2*(u+1)^2)/((u^2+3)^3*(u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31));
A4:=(64*(u^6+2*u^5+3*u^4-20*u^3+39*u^2+18*u+21)*(u-1)^6*(u+1)^6)/((u^2+3)^6*(u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31)^2);
A6:=(4096*(u-1)^12*(u+1)^12)/((u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31)^3*(u^2+3)^9);
E:=EllipticCurve([1,A2,0,A4,A6]);							// we create an elliptic curve given by the long Weierstrass model in section 2.4 of the paper D. Jeon, A. Schweizer "Torsion of rational elliptic curves over different types of cubic fields"

Factorization(Numerator(Discriminant(E)));						// gives the normalized factorization of the numerator of the discriminant
Factorization(Denominator(Discriminant(E)));						// gives the normalized factorization of the denominator of the discriminant
d_num_2:=Numerator(Discriminant(E));								// we have to check those because the factorization is normalized and we want to see whether there are any leading coefficients
d_den_2:=Denominator(Discriminant(E));
d_num_2;
d_den_2;

Factorization(Numerator(cInvariants(E)[1]));						// gives the normalized factorization of the numerator of the c_4-invariant
Factorization(Denominator(cInvariants(E)[1]));						// gives the normalized factorization of the denominator of the c_4-invariant
c4_num_2:=Numerator(cInvariants(E)[1]);								// we have to check those because the factorization is normalized and we want to see whether there are any leading coefficients
c4_den_2:=Denominator(cInvariants(E)[1]);	
c4_num_2;
c4_den_2;

Evaluate(d_num_2/(2^14*m^14), 0);							// this is odd, so d_num/(2^14*m^14) = 1 (mod 2)
Evaluate(d_den_2, 0);									// this is odd, so d_den =1 (mod 2)
Evaluate(c4_num_2, 0);									// this is odd, so c4_num =1 (mod 2)
Evaluate(c4_den_2, 0);									// this is odd, so c4_den =1 (mod 2)


aInvariants(E)[3];			// gives a_3, and we see that ord_2(a_3)>0
aInvariants(E)[4];			// gives a_4, and we see that ord_2(a_4)>0 (we have to look at the numerator)
aInvariants(E)[5];			// gives a_6, and we see that ord_2(a_6)>0 (we have to look at the numerator)
bInvariants(E)[1];			// gives b_2, and ord_2(b_2)=0, so the conditions of Lemma 2.1 are satisfied

aInvariants(E)[1];			// gives a_1=1
aInvariants(E)[2];			// gives a_2, and ord_2(a_2)>0 so we get the equation T^2+T=0 over F_2 in Lemma 2.1





//
// (3) ord_2(u)=0
//


K<m>:=PolynomialRing(Rationals());
u:=m+1;											// the substitution u-1-->m

A2:=(-4*(u^6+2*u^5+15*u^4-20*u^3+15*u^2+18*u+33)*(u-1)^2*(u+1)^2)/((u^2+3)^3*(u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31));
A4:=(64*(u^6+2*u^5+3*u^4-20*u^3+39*u^2+18*u+21)*(u-1)^6*(u+1)^6)/((u^2+3)^6*(u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31)^2);
A6:=(4096*(u-1)^12*(u+1)^12)/((u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31)^3*(u^2+3)^9);
E:=EllipticCurve([1,A2,0,A4,A6]);							// we create an elliptic curve given by the long Weierstrass model in section 2.4 of the paper D. Jeon, A. Schweizer "Torsion of rational elliptic curves over different types of cubic fields"

Factorization(Numerator(Discriminant(E)));						// gives the normalized factorization of the numerator of the discriminant
Factorization(Denominator(Discriminant(E)));						// gives the normalized factorization of the denominator of the discriminant
Numerator(Discriminant(E));								// we have to check those because the factorization is normalized and we want to see whether there are any leading coefficients
Denominator(Discriminant(E));

Factorization(Numerator(cInvariants(E)[1]));						// gives the normalized factorization of the numerator of the c_4-invariant
Factorization(Denominator(cInvariants(E)[1]));						// gives the normalized factorization of the denominator of the c_4-invariant
Numerator(cInvariants(E)[1]);								// we have to check those because the factorization is normalized and we want to see whether there are any leading coefficients
Denominator(cInvariants(E)[1]);	

aInvariants(E)[3];			// gives a_3, and we see that ord_2(a_3)>0
aInvariants(E)[4];			// gives a_4, and we see that for k>1 we have ord_2(a_4)>0 
aInvariants(E)[5];			// gives a_6, and we see that for k>1 we have ord_2(a_6)>0
bInvariants(E)[1];			// gives b_2, and ord_2(b_2)=0 (when we divide both numerator and the denominator with 2^12), so the conditions of Lemma 2.1 are satisfied

aInvariants(E)[1];			// gives a_1=1
aInvariants(E)[2];			// gives a_2, and for k>1 we have ord_2(a_2)>0 so we get the equation T^2+T=0 over F_2 in Lemma 2.1





//
// k=1
//

K<n>:=PolynomialRing(Rationals());
u:=2*n+1;											// the substitution u-->2n+1

A2:=(-4*(u^6+2*u^5+15*u^4-20*u^3+15*u^2+18*u+33)*(u-1)^2*(u+1)^2)/((u^2+3)^3*(u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31));
A4:=(64*(u^6+2*u^5+3*u^4-20*u^3+39*u^2+18*u+21)*(u-1)^6*(u+1)^6)/((u^2+3)^6*(u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31)^2);
A6:=(4096*(u-1)^12*(u+1)^12)/((u^6+4*u^5+13*u^4-40*u^3+19*u^2+36*u+31)^3*(u^2+3)^9);
E:=EllipticCurve([1,A2,0,A4,A6]);							// we create an elliptic curve given by the long Weierstrass model in section 2.4 of the paper D. Jeon, A. Schweizer "Torsion of rational elliptic curves over different types of cubic fields"

Factorization(Numerator(Discriminant(E)));						// gives the normalized factorization of the numerator of the discriminant
Factorization(Denominator(Discriminant(E)));						// gives the normalized factorization of the denominator of the discriminant
Numerator(Discriminant(E));								// we have to check those because the factorization is normalized and we want to see whether there are any leading coefficients
Denominator(Discriminant(E));

Factorization(Numerator(cInvariants(E)[1]));						// gives the normalized factorization of the numerator of the c_4-invariant
Factorization(Denominator(cInvariants(E)[1]));						// gives the normalized factorization of the denominator of the c_4-invariant
Numerator(cInvariants(E)[1]);								// we have to check those because the factorization is normalized and we want to see whether there are any leading coefficients
Denominator(cInvariants(E)[1]);	

aInvariants(E)[3];			// gives a_3, and we see that ord_2(a_3)>0
aInvariants(E)[4];			// gives a_4, and we see that ord_2(a_4)>0 (taking into consideration that ord_2(n)=0) 
aInvariants(E)[5];			// gives a_6, and we see that ord_2(a_6)>0 (taking into consideration that ord_2(n)=0) 
bInvariants(E)[1];			// gives b_2, and ord_2(b_2)=0 (taking into consideration that ord_2(n)=0), so the conditions of Lemma 2.1 are satisfied

aInvariants(E)[1];			// gives a_1=1
aInvariants(E)[2];			// gives a_2, and taking into consideration that ord_2(n)=0 we have ord_2(a_2)>0 so we get the equation T^2+T=0 over F_2 in Lemma 2.1