ellheightmatrix

Function: ellheightmatrix
Section: elliptic_curves
C-Name: ellheightmatrix
Prototype: GGp
Help: ellheightmatrix(E,x): gives the height matrix for vector of points x
 on elliptic curve E.
Doc: $x$ being a vector of points, this
 function outputs the Gram matrix of $x$ with respect to the N\'eron-Tate
 height, in other words, the $(i,j)$ component of the matrix is equal to
 \kbd{ellbil($E$,x[$i$],x[$j$])}. The rank of this matrix, at least in some
 approximate sense, gives the rank of the set of points, and if $x$ is a
 basis of the \idx{Mordell-Weil group} of $E$, its determinant is equal to
 the regulator of $E$. Note our height normalization follows Cremona's
 \emph{Algorithms for modular elliptic curves}: this matrix should be divided
 by 2 to be in accordance with, e.g., Silverman's normalizations.