Function: ellorder
Section: elliptic_curves
C-Name: ellorder
Prototype: GGDG
Help: ellorder(E,z,{o}): order of the point z on the elliptic curve E over
 a number field or a finite field, 0 if non-torsion. The parameter o,
 if present, represents a non-zero multiple of the order of z.
Doc: gives the order of the point $z$ on the elliptic
 curve $E$, defined over a finite field or a number field.
 Return (the impossible value) zero if the point has infinite order.
 \bprog
 ? E = ellinit([-157^2,0]);  \\ the "157-is-congruent" curve
 ? P = [2,2]; ellorder(E, P)
 %2 = 2
 ? P = ellheegner(E); ellorder(E, P) \\ infinite order
 %3 = 0
 ? K = nfinit(polcyclo(11,t)); E=ellinit("11a3", K); T = elltors(E);
 ? ellorder(E, T.gen[1])
 %5 = 25
 ? E = ellinit(ellfromj(ffgen(5^10)));
 ? ellcard(E)
 %7 = 9762580
 ? P = random(E); ellorder(E, P)
 %8 = 4881290
 ? p = 2^160+7; E = ellinit([1,2], p);
 ? N = ellcard(E)
 %9 = 1461501637330902918203686560289225285992592471152
 ? o = [N, factor(N)];
 ? for(i=1,100, ellorder(E,random(E)))
 time = 260 ms.
 @eprog
 The parameter $o$, is now mostly useless, and kept for backward
 compatibility. If present, it represents a non-zero multiple of the order
 of $z$, see \secref{se:DLfun}; the preferred format for this parameter is
 \kbd{[ord, factor(ord)]}, where \kbd{ord} is the cardinality of the curve.
 It is no longer needed since PARI is now able to compute it over large
 finite fields (was restricted to small prime fields at the time this feature
 was introduced), \emph{and} caches the result in $E$ so that it is computed
 and factored only once. Modifying the last example, we see that including
 this extra parameter provides no improvement:
 \bprog
 ? o = [N, factor(N)];
 ? for(i=1,100, ellorder(E,random(E),o))
 time = 260 ms.
 @eprog
Variant: The obsolete form \fun{GEN}{orderell}{GEN e, GEN z} should no longer be
 used.