//
//
// code used in Proposition 3.7
//
//





//
// N=14
//

j:=-3^3*5^3;
E:=EllipticCurveFromjInvariant(j);
E:=QuadraticTwist(E,-5*7);			// we are twisting by -5*7 to get a curve with the least conductor
CremonaReference(E);				// we get the curve 49a1=49.a4
LocalInformation(E);				// we have reduction type III at p=7


j:=3^3*5^3*17^3;
E:=EllipticCurveFromjInvariant(j);
E:=QuadraticTwist(E,-3*5*7*17*19);		// we are twisting by -3*5*7*17*19 to get a curve with the least conductor
CremonaReference(E);				// we get the curve 49a2=49.a3
LocalInformation(E);				// we have reduction type III at p=7





//
// N=17
//

j:=-(17^2*101^3)/2;
E:=EllipticCurveFromjInvariant(j);
E:=QuadraticTwist(E,5*101*7717);		// we are twisting by 5*101*7717 to get a curve with the least conductor
CremonaReference(E);				// we get the curve 14450p1=14450.b2
LocalInformation(E);				// we have reduction type III at p=5


j:=-(17*373^3)/2^17;
E:=EllipticCurveFromjInvariant(j);
E:=QuadraticTwist(E,-5*17*373*14891);		// we are twisting by -5*17*373*14891 to get a curve with the least conductor
CremonaReference(E);				// we get the curve 14450p2=14450.b1
LocalInformation(E);				// we have reduction type III at p=5





//
// N=19
//

j:=-2^15*3^3;
E:=EllipticCurveFromjInvariant(j);
E:=QuadraticTwist(E,19);			// we are twisting by 19 to get a curve with the least conductor
CremonaReference(E);				// we get the curve 361a1=361.a2
LocalInformation(E);				// we have reduction type III at p=19





//
// N=37
//

j:=-7*11^3;
E:=EllipticCurveFromjInvariant(j);
E:=QuadraticTwist(E,5*11*47);			// we are twisting by 5*11*47 to get a curve with the least conductor
CremonaReference(E);				// we get the curve 1225h1=1225.b2
LocalInformation(E);				// we have reduction type III at p=5


j:=-7*137^3*2083^3;
E:=EllipticCurveFromjInvariant(j);
E:=QuadraticTwist(E,-5*11*137*1433*2083*11443);	// we are twisting by -5*11*137*1433*2083*11443 to get a curve with the least conductor
CremonaReference(E);				// we get the curve 1225h2=1225.b1
LocalInformation(E);				// we have reduction type III at p=5





//
// N=43
//

j:=-2^18*3^3*5^3;
E:=EllipticCurveFromjInvariant(j);
E:=QuadraticTwist(E,-2*3*5*7*43);		// we are twisting by -2*3*5*7*43 to get a curve with the least conductor
CremonaReference(E);				// we get the curve 1849a1=1849.b2
LocalInformation(E);				// we have reduction type III at p=43





//
// N=67
//

j:=-2^15*3^3*5^3*11^3;
E:=EllipticCurveFromjInvariant(j);
E:=QuadraticTwist(E,5*7*11*31*67);		// we are twisting by 5*7*11*31*67 to get a curve with the least conductor
CremonaReference(E);				// we get the curve 4489a1=4489.b2
LocalInformation(E);				// we have reduction type III at p=67





//
// N=163
//

j:=-2^18*3^3*5^3*23^3*29^3;
E:=EllipticCurveFromjInvariant(j);
E:=QuadraticTwist(E,2*5*7*11*19*23*29*127*163);	// we are twisting by 2*5*7*11*19*23*29*127*163 to get a curve with the least conductor
CremonaReference(E);				// we get the curve 26569a1=26569.a2
LocalInformation(E);				// we have reduction type III at p=163