Function: ellmul
Section: elliptic_curves
C-Name: ellmul
Prototype: GGG
Help: ellmul(E,z,n): n times the point z on elliptic curve E (n in Z).
Doc:
computes $[n]z$, where $z$ is a point on the elliptic curve $E$. The
exponent $n$ is in $\Z$, or may be a complex quadratic integer if the curve $E$
has complex multiplication by $n$ (if not, an error message is issued).
\bprog
? Ei = ellinit([1,0]); z = [0,0];
? ellmul(Ei, z, 10)
%2 = [0] \\ unsurprising: z has order 2
? ellmul(Ei, z, I)
%3 = [0, 0] \\ Ei has complex multiplication by Z[i]
? ellmul(Ei, z, quadgen(-4))
%4 = [0, 0] \\ an alternative syntax for the same query
? Ej = ellinit([0,1]); z = [-1,0];
? ellmul(Ej, z, I)
*** at top-level: ellmul(Ej,z,I)
*** ^--------------
*** ellmul: not a complex multiplication in ellmul.
? ellmul(Ej, z, 1+quadgen(-3))
%6 = [1 - w, 0]
@eprog
The simple-minded algorithm for the CM case assumes that we are in
characteristic $0$, and that the quadratic order to which $n$ belongs has
small discriminant.