//
// code for finding the model y^2=f_26(x) for X_0(26) and the factorization of f_26 in Q(√13)[X]
//
_<x>:=PolynomialRing(Rationals());
C:=SimplifiedModel(SmallModularCurve(26));
C;
f:=x^6-8*x^5+8*x^4-18*x^3+8*x^2-8*x+1;
Factorization(f);
Discriminant(f);
Factorization(Integers()!Discriminant(f));

K<x>:=PolynomialRing(QuadraticField(13));
Factorization(f, K);





//
// code for finding the model y^2=f_28(x) for X_0(28) and the factorization of f_28 in Q(√-7)[X]
//
_<x>:=PolynomialRing(Rationals());
C:=SimplifiedModel(SmallModularCurve(28));
C;
f:=4*x^6-12*x^5+25*x^4-30*x^3+25*x^2-12*x+4;
Factorization(f);					// // this is the factorization of the normalized model, so we have to multiply it by 4, which is the leading coefficient of f
Discriminant(2*Factorization(f)[1][1]);
Factorization(Integers()!Discriminant(2*Factorization(f)[1][1]));
Discriminant(Factorization(f)[2][1]);
Factorization(Integers()!Discriminant(Factorization(f)[2][1]));
Discriminant(2*Factorization(f)[3][1]);
Factorization(Integers()!Discriminant(2*Factorization(f)[3][1]));

K<x>:=PolynomialRing(QuadraticField(-7));
Factorization(f, K);					





//
// code for finding the model y^2=f_29(x) for X_0(29) and the factorization of f_29 in Q(√29)[X]
//
_<x>:=PolynomialRing(Rationals());
C:=SimplifiedModel(SmallModularCurve(29));
C;
f:=x^6-4*x^5-12*x^4+2*x^3+8*x^2+8*x-7;
Factorization(f);
Discriminant(f);
Factorization(Integers()!Discriminant(f));

K<x>:=PolynomialRing(QuadraticField(29));
Factorization(f, K);





//
// code for finding the model y^2=f_30(x) for X_0(30) and the factorization of f_30 in Q(√5)[X]
//
_<x>:=PolynomialRing(Rationals());
C:=SimplifiedModel(SmallModularCurve(30));
C;
f:=x^8+14*x^7+79*x^6+242*x^5+441*x^4+484*x^3+316*x^2+112*x+16;
Factorization(f);
Discriminant(Factorization(f)[1][1]);
Factorization(Integers()!Discriminant(Factorization(f)[1][1]));
Discriminant(Factorization(f)[2][1]);
Factorization(Integers()!Discriminant(Factorization(f)[2][1]));
Discriminant(Factorization(f)[3][1]);
Factorization(Integers()!Discriminant(Factorization(f)[3][1]));

K<x>:=PolynomialRing(QuadraticField(5));
Factorization(f, K);





//
// code for finding the model y^2=f_33(x) for X_0(33) and the factorization of f_33 in Q(√-11)[X]
//
_<x>:=PolynomialRing(Rationals());
C:=SimplifiedModel(SmallModularCurve(33));
C;
f:=x^8+10*x^6-8*x^5+47*x^4-40*x^3+82*x^2-44*x+33;
Factorization(f);
Discriminant(Factorization(f)[1][1]);
Factorization(Integers()!Discriminant(Factorization(f)[1][1]));
Discriminant(Factorization(f)[2][1]);
Factorization(Integers()!Discriminant(Factorization(f)[2][1]));

K<x>:=PolynomialRing(QuadraticField(-11));
Factorization(f, K);





//
// code for finding the model y^2=f_35(x) for X_0(35) and the factorization of f_35 in Q(√5)[X]
//
_<x>:=PolynomialRing(Rationals());
C:=SimplifiedModel(SmallModularCurve(35));
C;
f:=x^8-4*x^7-6*x^6-4*x^5-9*x^4+4*x^3-6*x^2+4*x+1;
Factorization(f);
Discriminant(Factorization(f)[1][1]);
Factorization(Integers()!Discriminant(Factorization(f)[1][1]));
Discriminant(Factorization(f)[2][1]);
Factorization(Integers()!Discriminant(Factorization(f)[2][1]));

K<x>:=PolynomialRing(QuadraticField(5));
Factorization(f, K);





//
// code for finding the model y^2=f_39(x) for X_0(39) and the factorization of f_39 in Q(√13)[X]
//
_<x>:=PolynomialRing(Rationals());
C:=SimplifiedModel(SmallModularCurve(39));
C;
f:=x^8-6*x^7+3*x^6+12*x^5-23*x^4+12*x^3+3*x^2-6*x+1;
Factorization(f);
Discriminant(Factorization(f)[1][1]);
Factorization(Integers()!Discriminant(Factorization(f)[1][1]));
Discriminant(Factorization(f)[2][1]);
Factorization(Integers()!Discriminant(Factorization(f)[2][1]));

K<x>:=PolynomialRing(QuadraticField(13));
Factorization(f, K);





//
// code for finding the model y^2=f_40(x) for X_0(40) and the factorization of f_40 in Q(√-1)[X] and Q(√5)[X]
//
_<x>:=PolynomialRing(Rationals());
C:=SimplifiedModel(SmallModularCurve(40));
C;
f:=x^8+8*x^6-2*x^4+8*x^2+1;
Factorization(f);
Discriminant(f);
Factorization(Integers()!Discriminant(f));

K<x>:=PolynomialRing(QuadraticField(-1));
Factorization(f, K);

K<x>:=PolynomialRing(QuadraticField(5));
Factorization(f, K);





//
// code for finding the model y^2=f_41(x) for X_0(41) and the factorization of f_41 in Q(√41)[X]
//
_<x>:=PolynomialRing(Rationals());
C:=SimplifiedModel(SmallModularCurve(41));
C;
f:=x^8-4*x^7-8*x^6+10*x^5+20*x^4+8*x^3-15*x^2-20*x-8;
Factorization(f);
Discriminant(f);
Factorization(Integers()!Discriminant(f));

K<x>:=PolynomialRing(QuadraticField(41));
Factorization(f, K);





//
// code for finding the model y^2=f_48(x) for X_0(48) and the factorization of f_48 in Q(√-1)[X] and Q(√3)[X]
//
_<x>:=PolynomialRing(Rationals());
C:=SimplifiedModel(SmallModularCurve(48));
C;
f:=x^8+14*x^4+1;
Factorization(f);
Discriminant(Factorization(f)[1][1]);
Factorization(Integers()!Discriminant(Factorization(f)[1][1]));
Discriminant(Factorization(f)[2][1]);
Factorization(Integers()!Discriminant(Factorization(f)[2][1]));

K<x>:=PolynomialRing(QuadraticField(-1));
Factorization(f, K);

K<x>:=PolynomialRing(QuadraticField(3));
Factorization(f, K);





//
// code for finding the model y^2=f_50(x) for X_0(50) and the factorization of f_50 in Q(√5)[X]
//
_<x>:=PolynomialRing(Rationals());
C:=SimplifiedModel(SmallModularCurve(50));
C;
f:=x^6-4*x^5-10*x^3-4*x+1;
Factorization(f);
Discriminant(f);
Factorization(Integers()!Discriminant(f));

K<x>:=PolynomialRing(QuadraticField(5));
Factorization(f, K);