# rationalPointOnConic -- Rational point on a conic.

## Synopsis

• Usage:
rationalPointOnConic(I)
• Inputs:
• I, an ideal, the ideal of an irreducible conic.
• Optional inputs:
• vb => ..., default value 0, Option whether to print intermediate results.
• Outputs:
• , a row matrix over \mathbb{Q} with homogeneous integer coordinates of a rational point on the conic or 0-matrix if there is none.

## Description

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 = | 67 49 -70 | 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

## Caveat

Returns the 0-matrix if there is no rational point.