Suppose I have three rational points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ on certain elliptic curve. Then they are linearly independent if and only if the determinant of matrix $(<P_i,P_j>)_{i,j}$ is non zero where $<\_,\_ >$ is Neron-Tate height pairing.
How to write a code in Sage or Pari to compute this determinant. Note that I don't want to write an equation of my elliptic curve in the code to compute rational points first. Rather I know my rational points and want to directly put rational points and compute the determinant.
Best Answer
In SageMath this can be accomplished using the command
height_pairing_matrix
. For example:You can find more information, including many examples, in the documentation.