Function: ellheight
Section: elliptic_curves
C-Name: ellheight0
Prototype: GDGDGp
Help: ellheight(E,{P},{Q}): Faltings height of the curve E, resp. canonical
 height of the point P on elliptic curve E, resp. the value of the attached
 bilinear form at (P,Q).
Doc: Let $E$ be an elliptic curve defined over $K = \Q$ or a number field,
 as output by \kbd{ellinit}; it needs not be given by a minimal model
 although the computation will be faster if it is.

 \item Without arguments $P,Q$, returns the Faltings height of the curve $E$
 using Deligne normalization. For a rational curve, the normalization is such
 that the function returns \kbd{-(1/2)*log(ellminimalmodel(E).area)}.

 \item If the argument $P \in E(K)$ is present, returns the global
 N\'eron-Tate height $h(P)$ of the point, using the normalization in
 Cremona's \emph{Algorithms for modular elliptic curves}.

 \item If the argument $Q \in E(K)$ is also present, computes the value of
 the bilinear form $(h(P+Q)-h(P-Q)) / 4$.

 The canonical height is equal to the N\'eron-Tate height
 divided by the degree of the number field.
 For a curve over a number field, it is
 \kbd{ellheight(E,P$\{,Q\}$)/\#E.nf.zk}.

Variant: Also available is \fun{GEN}{ellheight}{GEN E, GEN P, long prec}
 ($Q$ omitted).
