Decides whether a conic has a rational point and if so computes one.
i1 : R=QQ[y_0..y_2]; |
i2 : I=ideal(7*y_0^2+11*y_2^2+13*y_0*y_2+17*y_1^2+19*y_1*y_2); o2 : Ideal of R |
i3 : p=rationalPointOnConic I o3 = | 1139 497 -1190 | 1 3 o3 : Matrix QQ <--- QQ |
i4 : sub(I,{y_0=>p_(0,0),y_1=>p_(0,1),y_2=>p_(0,2)}) o4 = ideal 0 o4 : Ideal of QQ |
i5 : I=ideal(y_0^2+y_1^2+2*y_0*y_1+y_2^2); o5 : Ideal of R |
i6 : p=rationalPointOnConic I o6 = | -1 1 0 | 1 3 o6 : Matrix QQ <--- QQ |
i7 : sub(I,{y_0=>p_(0,0),y_1=>p_(0,1),y_2=>p_(0,2)}) o7 = ideal 0 o7 : Ideal of QQ |
i8 : I=ideal(y_0^2+y_2^2+2*y_0*y_2+2*y_1^2+2*y_1*y_2+4*y_0*y_1); o8 : Ideal of R |
i9 : p=rationalPointOnConic I o9 = | 1 -1 1 | 1 3 o9 : Matrix QQ <--- QQ |
i10 : sub(I,{y_0=>p_(0,0),y_1=>p_(0,1),y_2=>p_(0,2)}) o10 = ideal 0 o10 : Ideal of QQ |
i11 : I=ideal(y_0^2+y_2^2+y_1^2); o11 : Ideal of R |
i12 : p=rationalPointOnConic I o12 = 0 1 3 o12 : Matrix QQ <--- QQ |